Franz Aurenhammer, Rolf Klein, Der-Tsai Lee, Voronoi Diagrams and Delaunay Triangulations

VORONOI DIAGRAMS AND
DELAUNAY TRIANGULATIONS
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VORONOI DIAGRAMS AND
DELAUNAY TRIANGULATIONS
Franz Aurenhammer
Graz University of Technology, Austria
Rolf Klein
University of Bonn, Germany
Der-Tsai Lee
Academia Sinica, Taiwan
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Aurenhammer, Franz, 1957–
Voronoi diagrams and Delaunay triangulations / Franz Aurenhammer, Graz University of
Technology, Austria, Rolf Klein, University of Bonn, Germany, Der-Tsai Lee, Academia Sinica,
Taiwan.
pages cm
Includes bibliographical references and index.
ISBN 978-9814447638 (hardcover : alk. paper)
1. Voronoi polygons. 2. Spatial analysis (Statistics) I. Klein, Rolf, 1953– II. Lee, Der-Tsai.
III. Title.
QA278.2.A97 2013
516.22--dc23
2013018154
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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CONTENTS
1. Introduction
1
2. Elementary Properties
7
2.1.
2.2.
Voronoi diagram . . . . . . . . . . . . . . . . . . . . . . . . . .
Delaunay triangulation . . . . . . . . . . . . . . . . . . . . . .
3. Basic Algorithms
3.1.
3.2.
3.3.
3.4.
3.5.
A lower time bound . .
Incremental construction
Divide & conquer . . .
Plane sweep . . . . . .
Lifting to 3-space . . .
15
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4. Advanced Properties
4.1.
4.2.
Characterization of Voronoi diagrams . . . . . . . . . . . . . .
Delaunay optimization properties . . . . . . . . . . . . . . . .
35
41
47
Line segment Voronoi diagram . . . . .
Convex polygons . . . . . . . . . . . . .
Straight skeletons . . . . . . . . . . . .
Constrained Delaunay and relatives . .
Voronoi diagrams for curved objects . .
5.5.1. Splitting the Voronoi edge graph
5.5.2. Medial axis algorithm . . . . . .
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6. Higher Dimensions
6.1.
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5. Generalized Sites
5.1.
5.2.
5.3.
5.4.
5.5.
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Voronoi and Delaunay tessellations in 3-space . . . . . . . . .
6.1.1. Structure and size . . . . . . . . . . . . . . . . . . . . .
v
75
75
vi
Contents
6.2.
6.3.
6.4.
6.5.
6.6.
6.1.2. Insertion algorithm . . . . . . . . . . .
6.1.3. Starting tetrahedron . . . . . . . . . .
Power diagrams . . . . . . . . . . . . . . . . .
6.2.1. Basic properties . . . . . . . . . . . . .
6.2.2. Polyhedra and convex hulls . . . . . .
6.2.3. Related diagrams . . . . . . . . . . . .
Regular simplicial complexes . . . . . . . . . .
6.3.1. Characterization . . . . . . . . . . . . .
6.3.2. Polytope representation in weight space
6.3.3. Flipping and lifting cell complexes . . .
Partitioning theorems . . . . . . . . . . . . . .
6.4.1. Least-squares clustering . . . . . . . .
6.4.2. Two algorithms . . . . . . . . . . . . .
6.4.3. More applications . . . . . . . . . . . .
Higher-order Voronoi diagrams . . . . . . . . .
6.5.1. Farthest-site diagram . . . . . . . . . .
6.5.2. Hyperplane arrangements and k-sets .
6.5.3. Computing a single diagram . . . . . .
6.5.4. Cluster Voronoi diagrams . . . . . . . .
Medial axis in three dimensions . . . . . . . .
6.6.1. Approximate construction . . . . . . .
6.6.2. Union of balls and weighted α-shapes .
6.6.3. Voronoi diagram for spheres . . . . . .
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7. General Spaces & Distances
7.1.
7.2.
7.3.
7.4.
7.5.
7.6.
Generalized spaces . . . . . . . . . . . . .
7.1.1. Voronoi diagrams on surfaces . .
7.1.2. Specially placed sites . . . . . . .
Convex distance functions . . . . . . . .
7.2.1. Convex distance Voronoi diagrams
7.2.2. Shape Delaunay tessellations . . .
7.2.3. Situation in 3-space . . . . . . . .
Nice metrics . . . . . . . . . . . . . . . .
7.3.1. The concept . . . . . . . . . . . .
7.3.2. Very nice metrics . . . . . . . . .
Weighted distance functions . . . . . . .
7.4.1. Additive weights . . . . . . . . . .
7.4.2. Multiplicative weights . . . . . . .
7.4.3. Modifications . . . . . . . . . . .
7.4.4. Anisotropic Voronoi diagrams . .
7.4.5. Quadratic-form distances . . . . .
Abstract Voronoi diagrams . . . . . . . .
7.5.1. Voronoi surfaces . . . . . . . . . .
7.5.2. Admissible bisector systems . . .
7.5.3. Algorithms and extensions . . . .
Time distances . . . . . . . . . . . . . . .
7.6.1. Weighted region problems . . . .
77
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123
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123
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128
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172
175
175
Contents
7.6.2.
7.6.3.
vii
City Voronoi diagram . . . . . . . . . . . . . . . . . . .
Algorithm and variants . . . . . . . . . . . . . . . . . .
8. Applications and Relatives
8.1.
8.2.
8.3.
8.4.
8.5.
183
Distance problems . . . . . . . . . . . . . . . . . .
8.1.1. Post office problem . . . . . . . . . . . . .
8.1.2. Nearest neighbors and the closest pair . .
8.1.3. Largest empty and smallest enclosing circle
Subgraphs of Delaunay triangulations . . . . . . .
8.2.1. Minimum spanning trees and cycles . . . .
8.2.2. α-shapes and shape recovery . . . . . . . .
8.2.3. β-skeletons and relatives . . . . . . . . . .
8.2.4. Paths and spanners . . . . . . . . . . . . .
Supergraphs of Delaunay triangulations . . . . . .
8.3.1. Higher-order Delaunay graphs . . . . . . .
8.3.2. Witness Delaunay graphs . . . . . . . . . .
Geometric clustering . . . . . . . . . . . . . . . . .
8.4.1. Partitional clustering . . . . . . . . . . . .
8.4.2. Hierarchical clustering . . . . . . . . . . .
Motion planning . . . . . . . . . . . . . . . . . . .
8.5.1. Retraction . . . . . . . . . . . . . . . . . .
8.5.2. Translating polygonal robots . . . . . . . .
8.5.3. Clearance and path length . . . . . . . . .
8.5.4. Roadmaps and corridors . . . . . . . . . .
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9. Miscellanea
9.1.
9.2.
9.3.
9.4.
Voronoi diagram of changing sites . .
9.1.1. Dynamization . . . . . . . . .
9.1.2. Kinetic Voronoi diagrams . . .
Voronoi region placement . . . . . . .
9.2.1. Maximizing a region . . . . .
9.2.2. Voronoi game . . . . . . . . .
9.2.3. Hotelling game . . . . . . . .
9.2.4. Separating regions . . . . . . .
Zone diagrams and relatives . . . . .
9.3.1. Zone diagram . . . . . . . . .
9.3.2. Territory diagram . . . . . . .
9.3.3. Root finding diagram . . . . .
9.3.4. Centroidal Voronoi diagram .
Proximity structures on graphs . . . .
9.4.1. Voronoi diagrams on graphs .
9.4.2. Delaunay structures for graphs
176
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225
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10. Alternative Solutions in Rd
10.1. Exponential lower size bound . . . . . . . . . . . . . . . . . . .
10.2. Embedding into low-dimensional space . . . . . . . . . . . . .
10.3. Well-separated pair decomposition . . . . . . . . . . . . . . . .
225
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254
viii
Contents
10.4. Post office revisited . . . . . . . . .
10.4.1. Exact solutions . . . . . . .
10.4.2. Approximate solutions . . .
10.5. Abstract simplicial complexes . . .
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11. Conclusions
11.1. Sparsely covered topics . . . . . . . . . . . . . . . . . . . . . .
11.2. Implementation issues . . . . . . . . . . . . . . . . . . . . . . .
11.3. Some open questions . . . . . . . . . . . . . . . . . . . . . . .
258
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261
267
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269
272
Bibliography
275
Index
329
Chapter 1
INTRODUCTION
This book is devoted to an important and influential geometrical structure
called the Voronoi diagram, whose origins in the Western literature date
back to at least the 17th century. In his work on the principles of
philosophy [257], René Descartes (Renatus Cartesius) claims that the solar
system consists of vortices. His illustrations show a decomposition of space
into convex regions, each consisting of matter revolving around one of the
fixed stars; see Figure 1.1.
Even though Descartes has not explicitly defined the extension of these
regions, the underlying idea seems to be the following. Let some space, and
a set of sites in this space be given, together with a notion of the influence
a site p exerts on every location x of the space. Then the region of the
site p consists of all points x for which the influence of p is the strongest.
Figure 1.2 illustrates the simplest case: Sites are points in the plane, and
influence is modeled by Euclidean distance. The regions of the sites are
convex polygons covering the entire plane.
‘The world is full of Voronoi diagrams’ . . . this phrase seems true
wherever we look, and it is the intention of this book to lead the reader
into the fascinating realm of Voronoi diagrams. Whether we look to the
sky, where gravitational fields structure the macrocosmos, or consider the
spread of animals in their habitats, simply watch the interference pattern
of waves on a quiet pond, or even peer deep within the structure of matter,
we will find a Voronoi diagram-like pattern which structures the world.
Indeed, this fundamental concept has emerged independently, and
proven useful, in various fields of science. Most applications have their own
notions of ‘space’, ‘sites’, and ‘influence’, resulting in Voronoi diagrams
whose structures differ greatly. Understanding their properties is the key
to their effective application and to the development of fast construction
algorithms. Various names have been given to Voronoi diagrams, depending
on the particular domain, such as Thiessen polygons in geography and
1
Chapter 1. Introduction
2
Figure 1.1.
Descartes’ decomposition of space into vortices.
meteorology (Thiessen [683], 1911), domains of action or Wirkungsbereiche
in crystallography (Niggli [562], 1927), Wigner–Seitz zones in chemistry and
physics (Wigner and Seitz [705], 1933), Johnson–Mehl model in mineralogy
(Johnson and Mehl [433], 1939), and medial axis transform in biology and
physiology (Blum [134], 1973).
Voronoi diagrams are important to both theory and applications, and
play a unique interdisciplinary role. Several thousands of research articles
have been published about them in different communities. Results thus
dispersed might not always be widely known, and might fade into oblivion.
While no single book can present all known results on Voronoi diagrams and
their relatives, our aim is to thoroughly cover the structural and algorithmic
viewpoints. In addition to being a versatile space partitioning structure,
Voronoi diagrams are also aesthetically pleasing, and many people feel
attracted to them, even regarding their artistic aspects. We have tried to
communicate this quality in this book.
Chapter 1. Introduction
3
p
Figure 1.2.
A Voronoi diagram of point sites in the Euclidean plane.
One and a half centuries ago, the mathematicians Carl Friedrich
Gauß [353] (1840) and Gustav Lejeune Dirichlet [280] (1850), and later
Georgi Feodosjewitsch Voronoi [693, 694] (1908) were the first to formally
introduce this concept. They used it to study quadratic forms: Here the
sites are integer lattice points, and influence is measured by Euclidean
distance. The resulting structure was called a Dirichlet tessellation or
Voronoi diagram, which became its standard name today.
Voronoi [693] also considered the geometrical dual of this structure,
where any two (point) sites are connected whose regions have a boundary in
common. Later, Boris Delaunay (Delone) [253] obtained the same structure
by defining that two sites are connected if they lie on a circle whose interior
contains no other sites. After him, the dual of the Voronoi diagram was
denoted Delaunay tessellation or Delaunay triangulation.
With the advent of modern computers, the important role of Voronoi
diagrams in structuring, representing, and displaying multidimensional
data was rediscovered by computational geometers. Voronoi diagrams are
a well-established geometric data structure nowadays. About one out of
sixteen articles in computational geometry is dedicated to them, ever since
Shamos and Hoey [637] introduced them to the field. In the two-dimensional
case, for instance in the Euclidean plane, the Voronoi diagram does not
require significantly more storage than does its underlying set of sites,
and thus captures the inherent proximity information in a comprehensive
and computationally useful manner. Its applications in more practically
oriented areas of computer science are numerous, for example, in geographic
4
Chapter 1. Introduction
information systems, robotics, computer graphics, and data classification
and clustering, to name a few.
Besides its direct applications in diverse fields of science, the Voronoi
diagram and its dual can be used for solving numerous, and surprisingly
different, geometric and graph-theoretical problems. Due to their close
relationship to polytopes and arrangements of hyperplanes in higher
dimensions, many questions (and solutions) from convex and discrete
geometry carry over to Voronoi diagrams. Moreover, the Delaunay
triangulation, seen as a combinatorial graph, is related to several prominent
connectivity graphs. We discuss various respective applications, often
including alternative solutions for their special merits. Along the way, we
give a state-of-the-art account of the literature on Voronoi diagrams in
computational geometry. This fills the need for a technically sound and
well-structured book, which is up-to-date on the theory and mathematical
applications of Voronoi diagrams.
The reader is invited on a guided tour of gently increasing difficulty
through a fascinating area. Insight will be given into the ideas and principles
of Voronoi diagrams, without the baggage of too much technical detail.
When later faced with a geometric partitioning problem, readers should
find our book helpful in deciding whether their problem shows Voronoi
characteristics and which type of Voronoi diagram applies. They might find
a ready-to-use solution in our book, or follow up the links to the literature
provided, or they might work out their own solution based on the algorithms
they have seen. The book targets researchers in mathematics, computer
science, natural and economical sciences, instructors and graduate students
in those fields, as well as the ambitious engineers looking for alternative
solutions. A brief discussion of algorithmic implementation questions, and
of currently available geometric computation libraries, is included in the
final chapter. Since human intuition is aided by visual perception, especially
where geometric topics are concerned, the diversity and appeal of Voronoi
diagrams is liberally illustrated with appropriate figures.
The presentation and structure of this book strives to highlight the
intrinsic potential of Voronoi diagrams and Delaunay triangulations, which
lies in their structural properties, in the existence of efficient algorithms
for their construction, and in their relationship to seemingly unrelated
concepts. We therefore organized topics by concept, rather than by
application.
Another book which nicely complements ours, but from a much
more applied perspective, is by Okabe et al. [571] (2000). It contains a
wealth of applications of Voronoi diagrams, several of them not (or only
marginally) covered here, like Poisson Voronoi diagrams and locational
Chapter 1. Introduction
5
optimization problems. The more than one thousand and six hundred
references listed in [571] constitute a bibliography still quite distinct from
ours — demonstrating once more the broad scope of Voronoi diagrams. In
addition, the book by Gavrilova [354] (2008) contains a collection of articles
on diverse applications in the natural sciences, with numerous citations. A
careful study of Delaunay triangulations, mainly oriented towards their
algorithmic applications in surface meshing and finite element methods,
is contained in George and Borouchaki [356] (1998). The recent book on
Delaunay mesh generation by Cheng, Dey, and Shewchuk [202] (2013) takes
into account the various new developments to date.
Shorter and early treatments of Voronoi diagrams and Delaunay
triangulations, closer to the spirit of the present book, are the surveys
by Aurenhammer [93], Fortune [343], and Aurenhammer and Klein [100].
Also, Chapters 5 and 6 of Preparata and Shamos [596], Chapter 13 of
Edelsbrunner [300], Chapters 7 and 9 of de Berg et al. [244], and Part V of
Boissonnat and Yvinec [151] could be consulted.
Although interesting and insightful in its own right, we decided
to refrain from a detailed historical treatment of Voronoi diagrams in
this book. The interested reader is encouraged to enjoy the historical
perspectives presented in [93], and later in more detail, in [571].
We start in Chapter 2 with a simple case — the Voronoi diagram
and the Delaunay triangulation of n points in the plane, under Euclidean
distance. We state elementary structural properties that follow directly
from the definitions. Further properties will be revealed in Chapter 3,
where different algorithmic schemes for computing these structures are
presented. In Chapter 4 we complete our presentation of the classical twodimensional case, with advanced properties of planar Voronoi diagrams and
Delaunay triangulations. We next turn to generalizations, to sites more
general than points in Chapter 5, and to higher dimensions in Chapter 6.
Generalized spaces and distances are treated elaborately in Chapter 7.
In Chapter 8, important geometric applications of the Voronoi diagram
and the Delaunay triangulation are discussed, along with respective related
structures and concepts. The reader interested mainly in these applications
can proceed directly to Chapter 8, after Chapter 2 or 3. Chapter 9
presents relevant topics which, for clarity of exposition, are best described
separately. Chapter 10 offers alternative solutions in high dimensions,
where the attractiveness of Voronoi diagrams is partially lost due to
their high combinatorial and computational worst-case complexity. Finally,
Chapter 11 concludes the book, gives a short discussion of algorithmic
implementation issues, and mentions some important open problems.
Chapter 2
ELEMENTARY PROPERTIES
In this chapter, we present definitions and basic properties of the Voronoi
diagram and its dual, the Delaunay triangulation.
Only the simplest case is considered — point sites in the plane under the
Euclidean distance. Yet, the properties discussed are of general importance,
as many (but not all) of them will carry over to other types of Voronoi
diagrams and their relatives, presented later in this book.
2.1. Voronoi diagram
Let us start with giving some standard notation and explanations.
Throughout this chapter, we will denote by S a set of n ≥ 3 point
sites p, q, r, . . . in the Euclidean plane, R2 . For points p = (p1 , p2 ) and
x = (x1 , x2 ), their Euclidean distance is given as
d(p, x) =
(p1 − x1 )2 + (p2 − x2 )2 .
The straight-line segment that connects two points p and q will be written
as pq, or sometimes just as pq.
For p, q ∈ S, let B(p, q) be the bisector of p and q (also called their
separator ), which is the locus of all points in R2 at equal distance from
both p and q. B(p, q) is the perpendicular line through the midpoint of the
line segment pq. It separates the halfplane
D(p, q) = {x | d(p, x) ≤ d(q, x)}
closer to p from the halfplane D(q, p) closer to q.
We next specify maybe the most important notion in this book. The
Voronoi region of p among the given set S of sites, for short VR(p, S), is
the intersection of the n − 1 halfplanes D(p, q), where q ranges over all the
7
8
Chapter 2. Elementary Properties
other sites in S. More formally,
VR(p, S) =
D(p, q).
q∈S, q=p
VR(p, S) consists of all points x ∈ R2 for which p is a nearest neighbor site.
As being the finite intersection of halfplanes, which are convex sets in R2 ,
the region VR(p, S) is a convex, possibly unbounded polygon in the plane.
Different Voronoi regions are interior-disjoint, as they are separated by the
bisector of the two sites that own them.
(A set M in R2 , or in general d-space Rd , is convex if it contains,
with each pair of points x, y ∈ M , the line segment xy. Set M is called
bounded if there exists a circle, or sphere, respectively, of finite radius which
encloses M . Otherwise, M is unbounded. We say that M is a closed set if M
contains its (topological) boundary. M minus its boundary is an open set,
called the interior of M .)
Definition 2.1. The common boundary part of two Voronoi regions is
called a Voronoi edge, if it contains more than one point.
The Voronoi diagram of S, for short V (S), is defined as the union of
all Voronoi edges.
Endpoints of Voronoi edges are called Voronoi vertices; they belong to
the common boundary of three or more Voronoi regions.
If a Voronoi edge e borders the regions of p and q then e ⊂ B(p, q)
holds. That is, V (S) is a planar straight-line graph whose edges emanate
from Voronoi vertices. V (S) is sometimes also referred to as the Voronoi
edge graph in the literature.
(Recall that a graph consists of a set of vertices, which in principle could
be any objects, and a set of edges that pair up certain vertices, in order
to display a relation between the two objects. A graph is called planar if
it can be geometrically embedded in the plane without edge crossings. For
sources on (combinatorial) graphs and graph-related algorithms, the books
by Diestel [275], Gibbons [363], or West [703] may be consulted.)
There is an intuitive way of looking at the Voronoi diagram V (S). Let
x be an arbitrary point in the plane. We center a circle, C, at x and let
its radius grow, from 0 on. At some stage the expanding circle will, for the
first time, hit one or more sites of S. Now there are three different cases.
Lemma 2.1. If the circle C expanding from point x hits exactly one site,
p, then x belongs to the interior of region VR(p, S). If C hits exactly two
sites, p and q, then x is an interior point of a Voronoi edge separating the
2.1. Voronoi diagram
9
regions of p and q. If C hits three or more sites simultaneously, then x is a
Voronoi vertex adjacent to those regions whose sites have been hit.
Proof. If only site p is hit then p is the unique element of S closest to x.
Consequently, x ∈ D(p, r) holds for each site r ∈ S with r = p. If C hits
exactly p and q, then x is contained in either halfplane D(p, r), D(q, r),
where r ∈ {p, q}, and also in B(p, q), the common boundary of D(p, q)
and D(q, p). By Definition 2.1, x belongs to the closure of the regions of
both p and q, but of no other site in S. In the third case, the argument is
analogous.
This lemma shows that the Voronoi diagram forms a decomposition (or
partition) of the plane; see Figure 2.1: Regions do not overlap, and for each
point x there is at least one closest site in S, meaning that x is covered by
this region.
Conversely, if we imagine n circles expanding from the sites at the
same speed, the fate of each point x of the plane is determined by those
sites whose circles reach x first. This ‘expanding waves’ view, or wavefront
model, has been systematically used by Chew and Drysdale [215] and
Thurston [685].
The Voronoi vertices are of degree at least three, by Lemma 2.1. (The
degree of a vertex is the number of its incident edges.) Vertices of degree
higher than three do not occur if no four point sites are cocircular. The
Γ
Figure 2.1.
Voronoi diagram of 11 sites and bounding curve Γ.
Chapter 2. Elementary Properties
10
Voronoi diagram V (S) is disconnected if all point sites are collinear; in this
case it consists of parallel lines.
From the Voronoi diagram of S one can easily derive the convex hull
of S, which is defined as the smallest convex set containing S. As S is finite,
its convex hull is a convex polygon, spanned by h ≤ n extreme points of S.
Lemma 2.2. A point p of S lies on the boundary of the convex hull of S
iff (i.e., if and only if ) its Voronoi region VR(p, S) is unbounded.
Proof. The Voronoi region of p is unbounded iff there exists some point
q ∈ S such that V (S) contains an unbounded piece of B(p, q) as a Voronoi
edge. Let x ∈ B(p, q), and let C(x) denote the circle through p and q
centered at x, as shown in Figure 2.2. Point x belongs to V (S) iff C(x)
contains no other site. As we move x to the right along B(p, q), the part of
C(x) contained in halfplane R keeps growing. If there is another site r in
R, it will eventually be reached by C(x), causing the Voronoi edge to end
at x. Otherwise, all other sites of S must be contained in the closure of the
left halfplane L. Then p and q both lie on the convex hull of S.
If the point set S is in convex position (i.e., all its points are vertices of
the convex hull of S) then all regions of V (S) are unbounded, by Lemma 2.2,
and their edges form a cycle-free and connected graph, that is to say, a tree.
Sometimes it is convenient to imagine a simple closed curve Γ around
the ‘interesting’ part of the Voronoi diagram, so large that it intersects only
the unbounded Voronoi edges; see Figure 2.1. (A curve is called simple if
it contains no self-intersections.) While walking along Γ, the vertices of the
L
R
p
C(x)
x
B(p,q)
q
l
Figure 2.2. As x moves to the right, the intersection of circle C(x) with the left
halfplane L shrinks, while C(x) ∩ R grows.
2.2. Delaunay triangulation
11
convex hull of S can be reported in cyclic order. After removing the halflines
outside Γ, a connected embedded planar graph with n + 1 faces results. Its
faces are the n Voronoi regions and the unbounded face outside Γ. We call
this graph the finite Voronoi diagram.
One virtue of the Voronoi diagram is its small size.
Lemma 2.3. The Voronoi diagram V (S) has O(n) many edges and
vertices. The average number of edges in the boundary of a Voronoi region
is less than 6.
Proof. By Euler’s polyhedron formula (see e.g. [363]) for planar graphs,
the following relation holds for the numbers v, e, f , and c of vertices, edges,
faces, and connected components, respectively
v − e + f = 1 + c.
We apply this formula to the finite Voronoi diagram. Each vertex has at
least three incident edges; by adding up we obtain e ≥ 3v/2, because each
edge is counted twice. Substituting this inequality together with c = 1 and
f = n + 1 yields
v ≤ 2n − 2
and e ≤ 3n − 3.
Adding up the numbers of edges contained in the boundaries of all n+1 faces
results in 2e ≤ 6n − 6, because each edge is again counted twice. Thus, the
average number of edges that border a region is at most (6n−6)/(n+1) < 6.
The same bounds apply to V (S).
2.2. Delaunay triangulation
Now we turn to the Delaunay tessellation. In general, a triangulation of S is
a planar graph with vertex set S and straight-line edges, which is maximal
in the sense that no further straight-line edge can be added without crossing
others. Triangulations are also called triangular networks in the literature.
Each triangulation of S contains the edges of the convex hull of S. Its
bounded faces are triangles, due to maximality. Their number equals exactly
2n − h − 2, where h counts the vertices of the convex hull. The number of
edges is 3n − h − 3. Note that both numbers, which easily follow from
Euler’s polyhedron formula, are independent from the way of triangulating
the point set S, by the characteristics of this formula.
We call a connected subset of edges of a triangulation a tessellation
of S if it contains the edges of the convex hull, and if each point of S has
at least two incident edges.
12
Chapter 2. Elementary Properties
Definition 2.2. The Delaunay tessellation DT(S) is obtained by
connecting with a line segment any two points p, q of S, for which a circle
exists that passes through p and q but does not contain any other site of S
in its interior or boundary. The edges of DT(S) are called Delaunay edges.
Equivalently, and also common in the literature, DT(S) can be defined
to contain all polygonal faces spanned by S whose circumcircles are
empty of points of S. This is the so-called empty-circle property of the
Delaunay tessellation. Another equivalent characterization of DT(S) is a
direct consequence of Lemma 2.1.
Lemma 2.4. Two points of S are joined by a Delaunay edge iff their
Voronoi regions are edge-adjacent.
Since each Voronoi region has at least two neighbors, at least two
Delaunay edges must emanate from each point of S. By the proof of
Lemma 2.2, each edge of the convex hull of S is Delaunay. Finally, two
Delaunay edges can only intersect at their endpoints, because they allow
for circumcircles whose respective closures do not contain other sites. This
shows that DT(S) is in fact a tessellation of S.
Two Voronoi regions can share at most one Voronoi edge, by convexity.
Therefore, Lemma 2.4 implies that DT(S) is the graph-theoretical dual
of V (S), realized by straight-line edges. Delaunay edges correspond to
Voronoi edges, Delaunay faces to Voronoi vertices (the centers of their
circumcircles), and Delaunay vertices (the n sites) to Voronoi regions —
in a bijective way.
An example is depicted in Figure 2.3; the Voronoi diagram V (S) is
drawn by dashed lines, and DT(S) by solid lines. Note that a Voronoi
vertex (like w) need not be contained in its associated face of DT(S). The
sites p, q, r, s are cocircular, giving rise to a Voronoi vertex v of degree 4.
Consequently, its corresponding Delaunay face is bordered by four edges.
This cannot happen if the points of S are in general position, i.e., no three
of them are collinear, and no four of them are cocircular.
Theorem 2.1. If the point set S is in general position then DT(S), the
dual of the Voronoi diagram V (S), is a triangulation of S, called the
Delaunay triangulation. Three points of S give rise to a Delaunay triangle
exactly if the circle they define does not enclose any other point of S.
Observe that DT(S) can be a triangulation even when S is not in
general position. Its number of triangles varies, as mentioned before, with
the number, h, of point sites which are extreme in S, i.e., which lie on the
boundary of the convex hull of S. A maximum of 2n − 5 is attained if the
2.2. Delaunay triangulation
13
V(S)
w
s
DT(S)
p
v
r
q
Figure 2.3.
Voronoi diagram and Delaunay tessellation.
convex hull is triangular (h = 3). On the other end of the spectrum, if S is
in convex position (h = n), then the convex hull is an n-gon, C, and DT(S)
partitions C into only n − 2 triangles, using diagonals of C.
As in every triangulation of a planar point set S, vertices of high
degree may occur in DT(S) for special positions of S. A site p ∈ S can
have n − 1 incident Delaunay edges (and triangles), because its Voronoi
region VR(p, S) can be adjacent to all other regions in V (S). Still, ‘almost
all’ sites will be of small constant degree in DT(S), because their average
degree is bounded by 6; see Lemma 2.3.
The Delaunay triangulation possesses various interesting features,
concerning its optimization properties and subgraph structures, but also its
flexibility in adapting to more general settings and higher dimensions. We
will discuss this later at appropriate places, mainly in Sections 4.2 and 8.2
and, regarding algorithmic issues, in Section 3.2 and Subsection 6.1.2.
We have seen, in this short chapter, two structures for a finite planar
point set S which are most fundamental and elementary, apart maybe
from the convex hull of S. The Voronoi diagram V (S) and the Delaunay
triangulation DT(S) display the proximity influence that S exerts in a
simple and intuitive way, with a description of only linear size. Their hidden
intrinsic potential will be gradually revealed in the chapters to come. We
start in Chapter 3, with showing that they lend themselves to several
efficient methods of construction.
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Chapter 3
BASIC ALGORITHMS
In this chapter we present several ways of computing a geometric
representation of the Voronoi diagram and its dual, the Delaunay
tessellation. For simplicity, we assume of the n point sites of S that no four
of them are cocircular, and that no three of them are collinear. According
to Theorem 2.1 we can then refer to DT(S) as the Delaunay triangulation.
All algorithms presented herein can be made to run without this general
position assumption. Also, they can be generalized to metrics other than
the Euclidean, and to sites other than points. This will be discussed at
appropriate places in later chapters.
Implementation issues, including degenerate configurations that might
occur in the input or during the execution of algorithms, along with general
numerical questions that arise in the computation of geometric objects,
will be discussed in some detail in Section 11.2. They are only marginally
addressed in the present chapter, not least for the sake of clarity in the
presentation of the algorithmic techniques and their analyses. For a rich
source on basic data structures (several of which will be used later on), we
refer to the books by Cormen et al. [234] and Mehlhorn and Sanders [538].
We mainly seek for a representation of Voronoi diagrams in exact vector
geometry, rather than as a binary image. Voronoi diagrams on pixel maps
can be obtained by exact geometric algorithms followed by pixel extraction,
or can be computed and visualized by graphics methods, beyond the scope
of this book; see Section 11.1.
Data structures well suited for working with planar graphs like the
Voronoi diagram are the doubly connected edge list (DCEL), by Muller and
Preparata [552], and the quad-edge structure by Guibas and Stolfi [394].
In either structure, a record is associated with each edge e that stores
the following information: the names of the two endpoints of e; references
to the edges clockwise or counterclockwise next to e about its endpoints;
15
Chapter 3. Basic Algorithms
16
finally, the names of the faces to the left and to the right of e. The space
requirement of both structures is O(n).
Either structure allows to efficiently traverse the edges incident to a
given vertex, or bordering a given face. The quad-edge structure offers the
additional advantage of describing, at the same time, a planar graph and
its dual, so that it can be used for constructing both the Voronoi diagram
and the Delaunay triangulation. From the DCEL of V (S) we can derive
the set of triangles constituting the Delaunay triangulation in linear time.
Conversely, from the set of all Delaunay triangles the DCEL of the Voronoi
diagram can be constructed in time O(n). Therefore, each algorithm for
computing one of the two structures can be used for computing the other
one, within O(n) extra time effort.
3.1. A lower time bound
Before constructing the Voronoi diagram we want to establish a lower bound
for its computational complexity.
Suppose that n real numbers x1 , . . . , xn are given, which are to be
sorted. We construct the point set S = {pi = (xi , xi 2 ) | 1 ≤ i ≤ n}, which
lies on the unit parabola and thus is in convex position; see Figure 3.1(i).
By Lemma 2.2, the Voronoi diagram V (S) forms a tree, from which one can
derive, in O(n) time, the vertices of the convex hull of S, in counterclockwise
order. From the leftmost point in S on, this vertex sequence contains all
points pi , sorted by increasing values of xi .
Y
Y
Y=X
2
p2
p3
yj
p5
x2
x5
X
x1
x6
pj
p4
p6
p1
pi
yi
x4
(i)
x3
X
iε
n
(ii)
Figure 3.1. Proving the Ω(n log n) lower bound for constructing the Voronoi diagram:
by transformation (i) from sorting, and (ii) from ε-closeness.
3.1. A lower time bound
17
This argument due to Shamos [636] shows that constructing the convex
hull and, a fortiori, computing the Voronoi diagram, is at least as difficult
as sorting n real numbers, which requires Ω(n log n) time in the algebraic
computation tree model [596].
However, a fine point is lost in this reduction. After sorting n points
by their x-values, their convex hull can be computed in linear time [286],
whereas sorting does not help in constructing the Voronoi diagram. The
following result has been independently found by Djidjev and Lingas [281]
and by Zhu and Mirzaian [716].
Theorem 3.1. It takes time Ω(n log n) to construct the Voronoi diagram
of n points p1 , . . . , pn whose x-coordinates are strictly increasing.
Proof. The proof is by reduction from the ε-closeness problem [596], which
is known to be in Ω(n log n). Let y1 , . . . , yn be positive real numbers, and
let ε > 0. The question is if there exists i = j such that |yi − yj | < ε holds.
We form the sequence of points
pi =
iε
, yi ,
n
for 1 ≤ i ≤ n,
and compute their Voronoi diagram; see Figure 3.1(ii). In time O(n), we
can determine the Voronoi regions that are intersected by the y-axis, in
bottom-up order (such techniques will be detailed in Section 3.3).
If, for each pi , its projection onto the y-axis lies in the Voronoi region
of pi then the values yi are available in sorted order, and we can easily
answer the question. Otherwise, there is a point pi whose projection lies in
the region of some other point pj . Because of
|yi − yj | ≤ d((0, yi ), pj ) < d((0, yi ), pi ) =
in this case the answer is positive.
iε
≤ ε,
n
On the other hand, sorting n arbitrary point sites by x-coordinates is
not made easier by their Voronoi diagram, as Seidel [626] has shown.
With Definition 2.1 in mind one could think of computing each Voronoi
region as the intersection of n − 1 halfplanes. This would take time
Θ(n log n) per region, see [596]. In the following sections we describe
various algorithms that compute the entire Voronoi diagram within this
time; due to Theorem 3.1, these algorithms are (asymptotically) worst-case
optimal.
18
Chapter 3. Basic Algorithms
3.2. Incremental construction
A natural idea first studied by Green and Sibson [378] is to construct
the Voronoi diagram by incremental insertion, i.e., to obtain V (S) from
V (S\{p}) by inserting the site p. In the beginning, when there are only two
sites, their Voronoi diagram is just their bisector line.
Basically, the cyclic sequence of edges of the new Voronoi region VR(p, S)
has to be constructed, and the invalidated parts of the diagram V (S\{p})
inside VR(p, S) deleted, in a way quite similar to constructing merge chains
of edges, as is detailed in the next section. Figure 3.2 illustrates the insertion
process. As the region of p can have up to n − 1 edges, for n = |S|, this leads
to a runtime of O(n2 ).
(Symbol |·| denotes the cardinality, when applied to finite sets, that is,
the number of elements in a set.)
Several authors fine-tuned the technique of inserting Voronoi regions,
and efficient and numerically robust implementations are available
nowadays; see Ohya et al. [570] and Sugihara and Iri [668]. In fact, overall
runtimes of O(n) can be expected for ‘well-distributed’ sets of sites.
The insertion process is, maybe, better described and implemented
in the dual environment, for the Delaunay triangulation: Construct the
triangulation DTi = DT({p1 , . . . , pi−1 , pi }) by inserting the site pi into
p
Figure 3.2. Inserting a Voronoi region. Invalidated portions of the diagram are drawn
in dashed style.
3.2. Incremental construction
19
DTi−1 . Clearly, DT3 is a single triangle, and DTn = DT(S) is the final
result. The advantage over a direct construction of V (S) is that Voronoi
vertices that appear in intermediate diagrams but not in the final one
need not be constructed and stored. We follow Guibas and Stolfi [394]
and construct DTi by exchanging edges, using Lawson’s [486] original edge
flipping procedure, until all edges invalidated by pi have been removed.
To this end, it is useful to extend the notion of triangle to the unbounded
face of the Delaunay triangulation, which is the complement of the convex
hull of S in the plane, R2 \conv(S). If pq is an edge of conv(S), we call the
supporting halfplane H not containing S an infinite triangle with edge pq.
Its ‘circumcircle’ is H itself, the limit of all circles through p and q whose
centers tend to infinity within H; cf. Figure 2.2. As a consequence, each
edge of a Delaunay triangulation is now incident to two triangles.
Those triangles of DTi−1 (finite or infinite) whose circumcircles contain
the new site, pi , are said to be in conflict with pi . According to Theorem 2.1,
they will no longer be Delaunay triangles.
Let qr be an edge of DTi−1 , and let T (q, r, t) be the triangle incident to
qr that lies on the other side of qr than pi ; see Figure 3.3. If its circumcircle
C(q, r, t) contains pi then each circle through q, r contains at least one of
pi and t; see Figure 3.3 again. Consequently, qr cannot belong to DTi , due
to Definition 2.2. Instead, pi t will be a new Delaunay edge, because there
exists a circle contained in C(q, r, t) that contains only pi and t in its interior
or boundary. This process of replacing edge qr by pi t is called an edge flip.
The necessary edge flips can be carried out efficiently if we know the
triangle T (q, s, r) of DTi−1 that contains pi , see Figure 3.4. (That is, we have
t
q
r
pi
C(pi,t)
C(q,r,t)
Figure 3.3. If triangle T (q, r, t) is in conflict with pi then the former Delaunay edge qr
must be replaced by pi t.
Chapter 3. Basic Algorithms
20
q
q
q
pi
pi
pi
T
F
s
t
t
t
s
r
r
SF
(i)
(ii)
s
r
SF
(iii)
Figure 3.4. Updating DTi−1 after inserting the new site pi . In (ii) the new Delaunay
edges connecting pi to q, r, s have been added, and edge qr has already been flipped.
Two more flips are necessary before the final state shown in (iii) is reached.
to perform point-location of pi in the temporary triangulation DTi−1 .) The
line segments connecting pi to q, r, and s will be new Delaunay edges, by
the same argument as described above. Next we check if, e.g., edge qr must
be flipped. If so, the edges qt and tr are tested, and so on. We continue until
no further edge currently forming a triangle with, but not containing pi ,
needs to be flipped, and obtain the triangulation DTi .
Lemma 3.1. If the triangle of DTi−1 containing pi is known, the
structural work needed for computing DTi from DTi−1 is proportional to
the degree m of pi in DTi .
Proof. Continued edge flipping replaces m − 2 conflicting triangles
of DTi−1 by m new triangles in DTi that have pi as a vertex; cf. Figure 3.4.
Lemma 3.1 yields an obvious O(n2 ) time algorithm for constructing the
Delaunay triangulation of n points: We can determine the triangle of DTi−1
containing pi within linear time, by inspecting all candidates. Moreover, the
degree of pi is trivially bounded by n − 1.
The last argument is quite crude. There can be single vertices in DTi
that do have a high degree, but their average degree is bounded by 6, as
Lemmas 2.3 and 2.4 show. This fact calls for randomization. Suppose we
pick pn at random in S, then choose pn−1 randomly from S − {pn }, and so
on. The result is a random permutation (p1 , p2 , . . . , pn ) of the set S of sites.
If we insert the sites in this order, each vertex of DTi has the same
chance of being pi . Consequently, the expected value of the degree of
pi is O(1), and the expected total number of structural changes in the
construction of DTn is only O(n), due to Lemma 3.1.
3.2. Incremental construction
21
In order to find the triangle that contains pi it is sufficient to inspect
all triangles that are in conflict with pi . The following lemma shows that
the expected total number of all conflicting triangles so far constructed is
only logarithmic.
Lemma 3.2. For each k < i, let tk denote the expected number of triangles
in DTk but not in DTk−1 that are in conflict with pi . Then,
i−1
tk = O(log i).
k=1
Proof. Let C denote the set of triangles of DTk that are in conflict
with pi . A triangle T ∈ C belongs to DTk \DTk−1 iff it has pk as a vertex.
As pk is randomly chosen in DTk , this happens with probability 3/k. Thus,
the expected number of triangles in C\DTk−1 equals 3 · |C|/k. Since the
expected size of C is less than 6 we have tk < 18/k, hence
i−1
tk < 18 ·
k=1
i−1
1/k = Θ(log i).
k=1
Suppose that T is a triangle of DTi incident to pi , see Figure 3.4(iii).
Its edge sr is in DTi−1 incident to two triangles: to its father, F , that has
been in conflict with pi ; and to its stepfather, SF, who is still present in
DTi . Any further site in conflict with T must be in conflict with its father
or with its stepfather, as illustrated by Figure 3.5.
F
pi
T
r
s
SF
Figure 3.5. The circumcirle of T is contained in the union of the circumcircles of its
father F and its stepfather SF.
22
Chapter 3. Basic Algorithms
This property can be exploited for quickly accessing all conflicting
triangles. The Delaunay tree due to Boissonnat and Teillaud [148] is a
directed acyclic graph that contains one node for each Delaunay triangle
ever created during the incremental construction. (In other words, this
graph reflects the partial order of the triangles imposed by the construction
history of the current Delaunay triangulation.) Pointers run from fathers
and stepfathers to their sons. The four triangles of DT3 (three of which are
infinite) are the sons of a dummy root node.
When pi must be inserted, a Delaunay tree including all triangles up to
DTi−1 is available. We start at its root and descend as long as the current
triangle is in conflict with pi . The above property guarantees that each
conflicting triangle of DTi−1 will be found.
The expected number of steps this search requires is only O(log i), due
to Lemma 3.2. Once DTi has been computed, the Delaunay tree can easily
be updated to include the new triangles. Thus, we have the following result.
Theorem 3.2. The Delaunay triangulation of a set of n points in the
plane can be constructed in expected time O(n log n), using expected linear
space. The average is taken over the different orders of inserting the n sites.
Note that we did not make any assumptions concerning the distribution
of the sites in the plane; the incremental algorithm achieves its O(n log n)
time bound for every possible input set. Only under a ‘poor’ insertion order
can a quadratic number of structural changes occur, but this is unlikely.
The user annoyed by the uncertainty hidden in an expected storage
requirement can apply a common trick to convert it into deterministic. Set
some parameter c, and let a memory manager terminate the execution —
and rerun the randomized construction — once the occupied space exceeds
c · n (see Lemma 2.3 for a suitable choice of c). The expected runtime
will still remain in O(n log n), though the constant in O will increase in
dependency of c.
As a nice feature, the insertion algorithm is online. That is, it is capable
of constructing DTi from DTi−1 without knowledge of pi+1 , . . . , pn . In fact,
site deletions can be handled similarly (as is sketched in Subsection 6.5.1,
and also in Subsection 6.5.3 as a special case of an on-line algorithm
which works in the more general setting of so-called higher-order Voronoi
diagrams). This allows for dynamizing Delaunay triangulations and Voronoi
diagrams, that is, maintaining these structures under insertion and deletion
series of sites. Dynamization can also be based on the divide & conquer
construction of Voronoi diagrams; see Section 3.3 for this earlier (and less
efficient) approach, and Section 9.1.
3.2. Incremental construction
23
Randomized geometric algorithms, though conceptually simple, tend
to be tricky to analyze. Since Clarkson and Shor [229] introduced their
technique, many researchers have been working on generalizing and
simplifying the methods used. To mention but a few results, Boissonnat
et al. [142] and Guibas et al. [390] have refined the methods of ‘storing
the past’ in order to locate new conflicts quickly, Clarkson et al. [228] and
Schwarzkopf [624] have generalized and simplified the analytic framework,
and Seidel [631] systematically applied the technique of backward analysis
first used by Chew [210]. A nice source is the monograph by Mulmuley [556].
The method in [390] for storing the past is applied in Section 6.5 for
constructing order-k planar Voronoi diagrams.
If the set S of sites can be expected to be well distributed in the
plane, bucketing techniques for accessing the triangle that contains a newly
inserted site pi have been used for speed-up. Joe [431], who implemented
Sloan’s algorithm [657], Su and Drysdale [664], who used a variant of
Bentley et al.’s spiral search [124], and Lemaire and Moreau [502], who also
gave probabilistic results for the higher-dimensional case, report on fast
experimental runtimes. Snoeyink and van Kreveld [660] describe a simple
way to precompute, in time O(n log n), an insertion order of the sites which
then guarantees a deterministic O(n)-time incremental construction of the
Delaunay triangulation — a task useful in the compression and transmission
of triangular networks.
The arising issues of numerical stability have been addressed in
Fortune [342], Sugihara [666], Jünger et al. [436], Shewchuk [648], and
Avnaim et al. [108]. The main geometric primitive used by the algorithm
is the incircle test, i.e., determining whether site pi is enclosed by the
circle defined by three other sites. As another practical issue, constructing
Delaunay triangulations for ‘imprecise point sets’ has been studied in
Buchin et al. [172] and papers cited therein. In this setting, the sites are
not known exactly but assumed to reside inside predefined regions (e.g.,
measurement tolerances), like small circles or squares.
Incremental insertion works well in the higher-dimensional case, and
also for certain types of generalized Voronoi diagrams and their duals. We
will see examples in Sections 6.1 and 6.5. Alternative methods for finding
a starting triangle, i.e., a triangle of the current triangulation that contains
the newly inserted site pi , are discussed in Devillers et al. [266] and are
reviewed, in the three-dimensional setting, in Section 6.1.
A technique similar to incremental insertion is incremental search. It
starts with a single Delaunay triangle, and then incrementally discovers new
ones, by growing triangles from edges of previously discovered triangles.
This basic idea is used, e.g., in Maus [524] and in Dwyer [296]. It leads
Chapter 3. Basic Algorithms
24
to efficient expected-time Delaunay algorithms, also in higher dimensions;
see [296].
The paper [664] gives a thorough experimental comparison of available
Delaunay triangulation algorithms.
3.3. Divide & conquer
The first deterministic worst-case optimal algorithm for computing the
Voronoi diagram has been presented by Shamos and Hoey [637]. In their
divide & conquer approach, the set of point sites, S, is split by a dividing line
into subsets L and R of about the same size. Then, the Voronoi diagrams
V (L) and V (R) are computed recursively, that is, the same strategy is
applied to the (smaller) point sets L and R. If only three or two points are
left in a set, their diagram is constructed directly, in O(1) time.
The essential part is in finding the split line, and in merging V (L) and
V (R), to obtain V (S). If these tasks can be carried out in time O(n) then
the overall running time, T (n), is only O(n log n), as we have the recurrence
relation
T (n) = 2 · T (n/2) + O(n).
During the recursion, vertical or horizontal split lines can be easily found
if the sites in S are sorted by their x- and y-coordinates beforehand.
The merge step involves computing the so-called merge chain B(L, R),
that is, the set of all Voronoi edges of V (S) that separate regions of sites
in L from regions of sites in R.
Suppose that the split line is vertical, and that L lies to its left.
Lemma 3.3. The edges of B(L, R) form a single y-monotone polygonal
chain. (That is, any line parallel to the x-axis intersects the chain in only
one point.) In V (S), the regions of all sites in L are to the left of B(L, R),
whereas the regions of the sites of R are to its right.
Proof. Let b be an arbitrary edge of B(L, R), and let l ∈ L and r ∈ R
be the two sites whose regions are adjacent to b. Since l has a smaller
x-coordinate than r, the edge b cannot be horizontal, and the region of l
must be to its left.
Thus, V (S) can be obtained by gluing together B(L, R), the part of
V (L) to the left of B(L, R), and the part of V (R) to its right; see Figure 3.6,
where V (R) is depicted by dashed lines.
The polygonal chain B(L, R) is constructed by finding a starting edge
at infinity, and by tracing B(L, R) through V (L) and V (R).
3.3. Divide & conquer
25
B(L,R)
V(L)
V(R)
Figure 3.6.
Merging V (L) and V (R) into V (S).
Due to Shamos and Hoey [637], an unbounded starting edge of B(L, R)
can be found in O(n) time by determining a line tangent to the convex
hulls of L and R, respectively. Here we describe an alternative method by
Lee [493], which was extended later by Chew and Drysdale [215], to be
applicable for generalized Voronoi diagrams (Section 7.2). The unbounded
regions of V (L) and V (R) are scanned simultaneously in cyclic order. For
each non-empty intersection VR(l, L) ∩ VR(r, R), we test if it contains an
unbounded piece of B(l, r). If so, this must be an edge of B(L, R), by
Definition 2.1. Since B(L, R) has two unbounded edges, by Lemma 3.3,
this search will be successful. It takes time |V (L)| + |V (R)| = O(n).
Now we describe how B(L, R) is traced. Suppose that the current edge b
of B(L, R) has just entered the region VR(l, L) at point v while running
within VR(r, R); see Figure 3.7. We determine the points vL and vR where
b leaves the regions of l and of r, respectively. The point vL is found by
scanning the boundary of VR(l, L) counterclockwise, starting from v. In
our example, vR is closer to v than vL , so that it must be the endpoint of
edge b.
From vR , B(L, R) continues with an edge b2 separating l and r2 . Now
we have to determine the points vL,2 and vR,2 where b2 hits the boundaries
of the regions of l and r2 . The crucial observation is that vL,2 cannot be
situated on the boundary segment of VR(l, L) from v to vL that we have just
scanned; this can be inferred from the convexity of VR(l, S). Therefore, we
Chapter 3. Basic Algorithms
26
B(l,r3)
B(l,r2)
B(l,r)
vL,2
l
VR(l,L)
b
r3
vR,2
b2
B(r2,r3)
vL
r2
vR
v
r
B(r,r2)
Figure 3.7.
Computing the merge chain B(L, R).
need to scan the boundary of VR(l, L) only from vL on, in counterclockwise
direction.
The same reasoning applies to V (R); only here, region boundaries are
scanned clockwise.
Even though the same region might be visited by B(L, R) several times,
no part of its boundary is scanned more than once. The edges of V (L) that
are scanned all lie to the right of B(L, R). This part of V (L), together
with B(L, R), forms a planar graph each of whose faces contains at least
one edge of B(L, R) in its boundary. As a consequence of Lemma 2.3, the
size of this graph does not exceed the size of B(L, R), times a constant.
The same holds for V (R). Therefore, the cost of constructing B(L, R) is
bounded by its size, once a starting edge is given. This leads to the following
result.
Theorem 3.3. The divide & conquer algorithm allows the Voronoi
diagram of n point sites in the plane to be constructed within time O(n log n)
and linear space, in the worst case. Both bounds are optimal.
Of course, the divide & conquer paradigm can also be applied to the
computation of the Delaunay triangulation DT(S). Guibas and Stolfi [394]
give an implementation that uses the quad-edge data structure and only two
3.3. Divide & conquer
27
geometric primitives: an orientation test and an incircle test. Fortune [342]
showed how to perform these tests accurately with finite precision.
Dwyer’s implementation [295] uses vertical and horizontal split lines
in turn, and Katajainen and Koppinen’s [447] merges square buckets in a
quad-tree order. Both papers report on favorable results.
Using the ‘history’ of constructing a Voronoi diagram by divide &
conquer, a dynamic Voronoi diagram algorithm can be designed. Gowda
et al. [376] pursue this approach. Based on a general dynamization paradigm
in Overmars [573], they propose the so-called Voronoi tree as a data
structure for supporting insertions and deletions of sites in O(n) time, when
n is the current size of the point set. The root of this binary tree stores
the diagram V (S) for the entire set S of sites, and its two children are the
roots of the Voronoi trees for the subsets L and R that S got split into.
Insertion or deletion of a site p amounts to an update in the Voronoi tree,
which can be accomplished by traversing a path between its root and the
leaf corresponding to p.
A related concept is the Voronoi diagram for moving sites, also called
the kinetic Voronoi diagram, where the diagram is to be updated during
a continuous movement of its sites along certain trajectories. We will
elaborate on this topic to some extent in Section 9.1.
Divide & conquer algorithms are candidates allowing for parallelization.
Efficient algorithms for computing in parallel the Voronoi diagram or the
Delaunay triangulation have been proposed. We refer to the paper by
Blelloch et al. [133] for references and for a practical parallel algorithm
for computing DT(S). They highlight an algorithm by Edelsbrunner and
Shi [314] that uses a lifting map for S (see Section 3.5) to construct a
chain of Delaunay edges that divides S. They show experimentally that
their implementation is comparable in work (i.e., product of runtime and
processor number) to the best sequential algorithms.
In all the divide & conquer approaches described in this section, the
emphasis is on the bottom-up phase — how to accomplish the merge step.
This step gets considerably complicated for Voronoi diagrams in more
general settings, because the merge chain may cycle and even heavily
disconnect.
An alternative approach, with emphasis on the top-down phase —
how to perform the divide step — has been recently considered in
Aichholzer et al. [34, 27]. It is based on constructing the medial axis of
a general planar shape, and exploits the tree structure of a medial axis
to calibrate the divide step. The conquer step is trivial and consists of
simply concatenating two partial medial axes. This algorithm, which also
28
Chapter 3. Basic Algorithms
works for quite general planar objects as sites, is described in detail in
Section 5.5.
3.4. Plane sweep
The well-known plane sweep algorithm (also called sweep line algorithm) by
Bentley and Ottmann [121] computes the intersections of n line segments
in the plane by moving a vertical line, H, from left to right across the plane.
The line segments currently intersected by H are stored in y-order. This
order must be updated whenever H reaches an endpoint of a line segment,
or an intersection point. To discover the intersection points in time, it is
sufficient to check, after each update of the y-order, those pairs of line
segments that have just become neighbors on H.
It is tempting to apply the same approach to Voronoi diagrams, by
keeping track of the Voronoi edges that are currently intersected by the
vertical sweep line. The problem is in discovering new Voronoi regions in
time. By the time the sweep line hits a new site, it has been intersecting
Voronoi edges of its region for a while.
Fortune [344] was the first to find a way around this difficulty. He
suggested a planar transformation under which each point site becomes
the leftmost point of its Voronoi region, so that it will be the first point
hit during a left-to-right sweep. His transformation does not change the
combinatorial structure of the Voronoi diagram.
Later, Seidel [629] and Cole [230] have shown how to avoid this
transformation altogether. They consider the Voronoi diagram of the point
sites to the left of the sweep line H and of H itself, considered an additional
site of straight-line shape; see Figure 3.8. Because the bisector of a line and
a non-incident point is a parabola, the boundary of the Voronoi region of
H is a connected chain of parabolic segments whose top- and bottommost
edges tend to infinity. This chain is called the wavefront, W .
Let p be a point site to the left of H. Any point to the left of, or on, the
parabola B(p, H) is not farther from p than from H; hence, it is a fortiori
closer to p than to any site to the right of H. Consequently, as the sweep
line moves on to the right, the waves must follow because the sets to the
left of B(pi , H) grow. On the other hand, each Voronoi edge to the left of
W that currently separates the regions of two sites pi , pj will be (part of)
a Voronoi edge in V (S).
During the sweep, there are two types of events that cause the structure
of the wavefront to change, namely when a new wave appears in W , or when
an old wave disappears. The former one, called a site event, happens each
time the sweep line hits a new site, e.g., p6 in Figure 3.8. At that very
3.4. Plane sweep
29
W’
W
p6
p4
p6
p4
p2
v’
p2
v
v
p3
p1
p3
p5
p5
p1
H
H’
Figure 3.8. Voronoi diagrams of the sweep line H, and of the points to its left, which
have been swept over already.
moment, B(H, p6 ) is a horizontal line through p6 (as a degenerate case).
A little later, its left halfline unfolds into a parabola that must be inserted
into the wavefront by gluing it onto the wave of p4 (which thereby is split
into two waves of W .)
For describing the other type of event, let p, q be two point sites whose
waves are neighbors in W . Their bisector, B(p, q), gives rise to a Voronoi
edge to the left of W . Its prolongation into the region of H is called a spike.
In Figure 3.8 spikes are depicted as dashed lines; one can think of them
as tracks along which the waves are moving. A wave disappears from W
when it arrives at the point where its two bounding spikes intersect. Its
former neighbors become now adjacent in the wavefront. This is called a
spike event.
In Figure 3.8, the wave of p3 would disappear at intersection point v,
if the new site, p6 , that causes a site event prior to that did not exist. But
after the wave of p6 has been inserted, a spike event at point v will take
place, and the corresponding wave of p4 will disappear.
While keeping track of the wavefront one can easily maintain the
portion of the Voronoi diagram to the left of W . Spikes correspond to
30
Chapter 3. Basic Algorithms
Voronoi edges. At each event, a new adjacency between two waves occurs,
and with it, a new spike. That is, a new Voronoi edge has been identified. In
particular, a Voronoi vertex gets constructed at each spike event. As soon
as all point sites have been detected and all spike intersections have been
processed, V (S) is obtained by removing the wavefront and extending to
infinity all spikes that have been intersecting it.
Even though one site may contribute several waves to the wavefront,
the following holds.
Lemma 3.4. The size of the wavefront is O(n).
Proof. Since any two parabolic bisectors B(p, H), B(q, H) can cross at
most twice, the number of waves of the wavefront is bounded by λ2 (n) =
2n − 1, where λs (n) denotes the maximum length of a Davenport–Schinzel
sequence over n symbols in which no two symbols appear s times each in
alternating positions; see [81].
The wavefront can be implemented by a balanced search tree that stores
the parabola segments in y-order. This enables us to insert a wave, or
remove a wave segment, in time O(log n).
Before the sweep starts, the point sites are sorted by increasing
x-coordinates and inserted into an event priority queue. (See e.g. [234]
or [538] for a description of basic data structures.) After each update of
the wavefront, the newly adjacent spikes are tested for intersection. If they
intersect at some point v, we insert into the event queue the time (i.e.,
the position x of the sweep line) when the wave segment between the two
spikes arrives at v. Since the point v is a Voronoi vertex of V (S), there are
only O(n) events caused by spike intersections. In addition, each of the n
sites causes an event. For each active spike we need to store only its first
intersection event. Thus, the size of the event queue never exceeds O(n).
We obtain the following result.
Theorem 3.4. Plane sweep provides an alternative way of computing the
Voronoi diagram of n points in the plane within O(n log n) time and O(n)
space.
McAllister et al. [527] have pointed out a subtle difference between
the sweep technique and the two methods mentioned before. The divide &
conquer algorithm computes Θ(n log n) many vertices, even though only
a linear number of them appears in the final diagram. The randomized
incremental construction method performs an expected Θ(n log n) number
of conflict tests. Both tasks, constructing a Voronoi vertex and testing a
subset of sites for conflict, are usually handled by subroutines that deal
3.5. Lifting to 3-space
31
directly with point coordinates, bisector equations, etc. They can become
quite costly if we consider sites more general than points, and distance
measures more general than the Euclidean distance; see Chapters 5 and 7.
The sweep algorithm, on the other hand, processes O(n) spike events,
and thus constructs only that many vertices.
The data structure to store the Voronoi diagram can be any standard
data structure for planar subdivisions. See, for example, the doubly
connected edge list [552] or the quad-edge data structure [394].
Note finally that the plane-sweep technique reduces a two-dimensional
‘static’ problem (for instance, the construction of a Voronoi diagram in the
plane) to a one-dimensional ‘dynamic’ problem (here, the maintenance of
the wavefront representation on the sweep line). This enables us to use
efficient one-dimensional data structures, like balanced binary trees and
priority queues. A kind of opposite philosophy underlies the algorithmic
technique presented in the subsequent section: An interpretation of the
planar problem in 3-space is sought, in order to gain insight into
its structural properties, and to apply known but seemingly unrelated
algorithms.
3.5. Lifting to 3-space
The following approach employs the powerful method of geometric
transformation, which leads to an optimal algorithm for constructing the
planar Delaunay triangulation.
Let P = {(x1 , x2 , x3 ) | x21 + x22 = x3 } denote the paraboloid depicted in
Figure 3.9. For each point x = (x1 , x2 ) in the plane, let x = (x1 , x2 , x21 +x22 )
denote its lifted image on P .
Z
x3=x21+x22
E
C’
p’
r’
q’
r
q
p
C
Figure 3.9.
Lifting circles onto the paraboloid.
32
Chapter 3. Basic Algorithms
Lemma 3.5. Let C be a circle in the plane. Then C is a planar curve on
the paraboloid P .
Proof. Suppose that C is given by the equation
r2 = (x1 − c1 )2 + (x2 − c2 )2 = x21 + x22 − 2x1 c1 − 2x2 c2 + c21 + c22 .
By substituting x21 + x22 = x3 we obtain
x3 − 2x1 c1 − 2x2 c2 + c21 + c22 − r2 = 0
for the points of C . This equation defines a plane in 3-space.
This lemma has an interesting consequence. Let S be a finite point set in
the plane, and denote by S its lifted image on P . The convex hull of S is a
convex polyhedron in 3-space. All points of S appear as polyhedron vertices,
because S is in convex position, by construction. (Compare Figure 3.1 (i)
in Section 3.1 for a picture in one dimension less.) By the lower convex hull
of S we mean that part of the convex hull which is visible from x3 = −∞.
Theorem 3.5. The Delaunay triangulation DT(S) of S equals the vertical
projection onto the x1 x2 -plane of the lower convex hull of S .
Proof. Let p, q, r denote three point sites of S. By Lemma 3.5, the lifted
image, C , of their circumcircle C lies on a plane, E, that cannot be vertical.
Under the lifting mapping, the points inside C correspond to the points on
the paraboloid P that lie below the plane E.
By Theorem 2.1, p, q, r define a triangle of the Delaunay triangulation
iff their circumcircle contains no further site. Equivalently, no lifted site
s is below the plane E that passes through p , q , r . But this means that
p , q , r define a facet of the lower convex hull of S .
Because there exist O(n log n) time algorithms for computing the
convex hull of n points in 3-space — see e.g. Preparata and Shamos [596],
and for a survey of the various available construction methods — see
Seidel [633], we have obtained another optimal algorithm for the Voronoi
diagram.
The connection between Voronoi diagrams and convex hulls has first
been studied by Brown [169] who used the inversion transform. The simpler
lifting mapping has been used, e.g., in Edelsbrunner and Seidel [310]. We
shall see several applications and generalizations in Chapter 6. In [310] also
the following fact is observed.
For each point s = (s1 , s2 ) of S, consider the paraboloid
Ps = {(x1 , x2 , x3 ) | (x1 − s1 )2 + (x2 − s2 )2 = x3 }.
3.5. Lifting to 3-space
33
If these paraboloids were opaque, and of pairwise different colors, an
observer looking from x3 = −∞ upwards would see the Voronoi diagram
V (S). In fact, for p, q ∈ S, the projection x = (x1 , x2 ) of a point
(x1 , x2 , x3 ) ∈ Pp ∩ Pq belongs to their bisector B(p, q); and there is no
site s closer to x than p and q iff (x1 , x2 , x3 ) lies below all paraboloids Ps .
That is, V (S) is the vertical projection onto the plane of the lower envelope
of these paraboloids.
Instead of the paraboloids Ps one could use the surfaces
{(x1 , x2 , x3 ) = f ((x1 − s1 )2 + (x2 − s2 )2 )}
generated by any function f that is strictly increasing. For example,
√
f (x) = x gives rise to cones of slope 45◦ with apices at the sites. This
setting illustrates the concept of circles expanding from the sites at equal
speed, as mentioned after the proof of Lemma 2.1. Coordinate x3 represents
time.
In order to visualize a Voronoi diagram on a graphic screen one can
feed the n surfaces to a z-buffer, and eliminate by brute force those parts
not visible from below; cf. Section 11.1 for so-called pixel Voronoi diagrams.
Finally, we would like to mention a nice connection between the
two ways of obtaining the Voronoi diagram by means of lower envelopes
of paraboloids explained above; it goes back to [310]. For a point
w = (w1 , w2 , w3 ), let ŵ denote its mirror image (w1 , w2 , −w3 ). If we apply
to 3-space the mapping which sends
x = (x1 , x2 , x3 ) to (x1 , x2 , x3 − (x1 − s1 )2 − (x2 − s2 )2 )
then, for each point s in the plane, the paraboloid Ps corresponds to the
tangent plane of the paraboloid P̂ at the point ŝ , where s denotes the lifted
image of s; compare the plane equation derived in the proof of Lemma 3.5,
letting c = s and r = 0.
Lower envelope representations of generalized Voronoi diagrams will be
briefly treated in Section 7.5. The corresponding surfaces and the resulting
diagram, respectively, are sometimes called Voronoi surfaces and their
minimization diagram in the literature. They are very useful for obtaining
algorithms and, in addition, bounds on the combinatorial complexity of
Voronoi diagrams.
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Chapter 4
ADVANCED PROPERTIES
We have seen, in Chapters 2 and 3, quite a few basic properties of
the Voronoi diagram for point sites in the plane. Many more shall be
described in later chapters, along with generalizations of the Voronoi
diagram concerning its sites and its underlying distance function, as certain
properties extend to such settings in a natural way. The present chapter is
devoted to some advanced features of the classical Voronoi diagram and its
dual, the Delaunay triangulation.
4.1. Characterization of Voronoi diagrams
The process of constructing the Voronoi diagram for n point sites can
be seen as an assignment of a planar convex region to each of the sites,
according to the ‘nearest neighbor rule’. We now address the following, in
some sense inverse, question: Given a partition of the plane into n convex
regions (which are then necessarily polygonal), do there exist sites, one for
each region, such that the nearest neighbor rule is fulfilled? In other words,
when is a given convex partition the Voronoi diagram of some set of sites?
Whether a given set of sites induces a given convex partition as its
Voronoi diagram is, of course, easy to decide by exploiting symmetry
properties among the sites. For the same reason, it is easy to check whether
a given triangulation is Delaunay, by exploiting the empty-circle property
of its triangles, stated in Theorem 2.1. Conditions for a given graph to be
isomorphic to the Delaunay triangulation of some set of sites are mentioned,
e.g., in the survey article by Fortune [343]. Below we concentrate on the
recognition of Voronoi diagrams without knowing the sites. The process of
restoring the sites, if they exist, is also called Voronoi diagram inversion;
see Schoenberg et al. [619] for an overview.
Questions of this kind arise in facility location and in the recognition
of biological growth models (as reported, e.g., in Suzuki and Iri [671]) and,
35
36
Figure 4.1.
Chapter 4. Advanced Properties
A Voronoi diagram?
Figure 4.2.
This one, yes.
in particular, in the so-called gerrymander problem mentioned in Ash and
Bolker [79]: When the sites are regarded as polling places and election
law requires each person to vote at the respective closest polling place,
the election districts form a Voronoi diagram. If the legislature draws
the district lines first, how can we tell whether election law is satisfied?
(Figures 4.1 and 4.2).
Let Ri and Rj be two of the given regions. Assume that they share a
common edge, and let hij be the straight line containing that edge. Further,
let σij denote the reflection at line hij .
Lemma 4.1. A convex partition R1 , . . . , Rn of the plane defines a Voronoi
diagram if and only if there exists a point pi for each region Ri such that
(1) pi ∈ Ri (containment condition),
(2) σij (pi ) = pj if Rj is adjacent to Ri (reflection condition).
Proof. If we do have a Voronoi diagram then its defining sites exist and
obviously fulfill (1) and (2). To prove the converse, assume that points
p1 , . . . , pn fulfilling both conditions exist. Take any region Ri and any
point x therein. We show that d(x, pi ) is minimum.
To get a contradiction, suppose pj , j = i, is closest to x. Consider an
edge of Rj that is intersected by the line segment xpj , and let Rk be the
region adjacent to Rj at that edge; see Figure 4.3. Note that k = i may
happen. By convexity of Rj and by (1), the line hjk separates pj from x.
Hence, by (2), we get d(x, pk ) < d(x, pj ), which is a contradiction.
We conclude that pi is closest to x among p1 , . . . , pn , which implies that
Ri is the region of pi in the Voronoi diagram V ({p1 , . . . , pn }).
4.1. Characterization of Voronoi diagrams
37
h jk
Rj
Ri
x
pj
p
k
Figure 4.3.
Region Ri must be closest to point x.
Based on Lemma 4.1, the recognition problem can now be formulated
as a linear programming problem; see Hartvigsen [398]. We first exploit the
reflection condition to get a system of linear equations.
Reflection at a line is an affine transformation, so we may write σij (x)
as Aij x + bij , for an appropriate matrix Aij and vector bij . Consider a
depth-first search order (see e.g. [363]) of the regions, in a way such that for
each region Ri an adjacent region Ri+1 is known. To get a linear system in
x, put
p1 = x,
p2 = A12 x + b12 := C2 x + d2 ,
p3 = A23 (A12 x + b12 ) + b23 := C3 x + d3
and so on. This expresses all points pi in terms of p1 by using n − 1
adjacencies among the regions. Each of the remaining adjacencies now gives
an equation of the form
Aij (Ci x + di ) + bij = Cj x + dj .
This system has at most 3n−3−(n−1) = 2n−2 equations by Lemma 2.3. If
it has no solution, or a unique solution, then we are done. In the former case,
we cannot have a Voronoi diagram. In the latter, we get the coordinates of
the first candidate site p1 = x. The corresponding other sites are obtained
simply by reflection. It remains to test these sites for containment in their
regions.
Chapter 4. Advanced Properties
38
Figure 4.4.
Sites might be chosen within the polygonal areas.
Setting up the system, solving it, and testing for containment can
be accomplished in time O(n) by standard methods. Note that only the
equations of the lines bounding the regions and the adjacency information
among the regions are needed. No coordinates of the region vertices are
required. This is particularly interesting for the recognition problem in
higher dimensions, to which the method above generalizes naturally.
The solution space of the linear system above may have dimension 1
or 2. Figure 4.4 reveals that certain symmetries among the regions lead
to non-unique solutions. Now the containment condition is exploited to
get, in addition, a set of inequalities for x. Consider each region Ri as the
intersection of all halfplanes bounded by the lines hij . Then pi ∈ Ri gives
a set of inequalities of the form
pTi tij ≤ aij ,
which, by plugging in pi = Ci x + di , yields
(CiT tij )x ≤ aij − dTi tij .
Finding a feasible solution of the corresponding linear program means
finding a possible site p1 = x for R1 which, by reflection, gives all the desired
sites. Since we deal with a linear program with O(n) constraints and of
4.1. Characterization of Voronoi diagrams
39
constant dimension (actually two), also only linear time (Megiddo [531]) is
spent in this more complicated case.
Theorem 4.1. Let C be a partition of the plane into n convex regions,
given by halfplanes supporting the regions and by adjacencies among regions.
O(n) time suffices for deciding whether C is a Voronoi diagram, and also
for restoring a suitable set of sites in case of its existence.
This result is clearly optimal, and the underlying method easy to
program. A generalization to higher dimensions is straightforward. Still,
the method has to be used with care, as even a slight deviation from
the correct Voronoi diagram (stemming from imprecise measurement or
numerical errors) will cause the method to classify C as non-Voronoi. Suzuki
and Iri [671] give a completely different method capable of approximating
C by a Voronoi diagram.
Lemma 4.1 extends to more general Voronoi-like partitions. The
characterizing configuration of points is then called a reciprocal figure,
valuing Clerk Maxwell’s work [525] of 1864. A nice survey on this subject
is Ash et al. [80]. In particular, if the partition C is polygonal, then the
containment and reflection conditions can be relaxed to
(1b) orthogonality: if polygons Ri and Rj share an edge e then the line
segment pi pj is orthogonal to e, and
(2b) orientation: any ray which is parallel to the vector (pj − pi ) and
intersects e meets Ri first.
Interestingly, the existence of a reciprocal figure {p1 , . . . , pn } for C, which is
now also called an orthogonal dual, is equivalent to the property that C can
be realized as the equilibrium state of a spider web: The edges of C allow
positive tensions that balance out at all vertices of C — a property of use
in the statics of plane frameworks; see e.g. Crapo and Whiteley [235].
Consult Figure 4.5. Suitable tensions for the edges eij = Ri ∩ Rj are
the lengths of the vectors (pj − pi ), or constant multiples thereof. These
tensions trivially add up to zero for the regions R1 , . . . , Rk around each
vertex v, because the corresponding line segments pi pi+1 border a face dual
to v (a triangle in the non-degenerate case).
Moreover, by Maxwell’s theorem [525], the existence of such tensions
characterizes partitions of the plane that can be obtained as the boundary
projections of convex polyhedra in 3-space; see also [235].
Clearly, for the Voronoi diagram V (S) its set S of sites is a possible
orthogonal dual. On the other hand, the vertices of V (S) constitute an
orthogonal dual for the Delaunay triangulation DT(S); the nonconvex
Chapter 4. Advanced Properties
40
70
52
63
42
28
35
64
p
42
2
30
48
p
39
39
3
58
23
40
67
v
p
1
93
57
Figure 4.5. Spider web with orthogonal dual (•). Any scaled and translated copy is a
valid orthogonal dual, too. Numbers at edges express tensions in equilibrium state at
each vertex. This is witnessed for vertex v by the side lengths of its dual triangle p1 p2 p3 .
unbounded face of DT(S) can be treated consistently. Indeed, in Section 3.5
we have seen the existence of a projection polyhedron for DT(S), namely,
the convex hull of the point set S lifted to R3 .
The correspondences above still hold in higher dimensions; see
Aurenhammer [88, 89]. By the relationship of Voronoi diagrams to convex
polyhedra in the next dimension (Subsection 6.2.2), a partition C admits
an orthogonal dual if and only if C is the power diagram of some set of
spheres; the dual points pi serve as sphere centers.
Various types of convex cell complexes can be identified as polyhedral
projections in this way, including simple complexes, arrangements of
hyperplanes, and higher-order Voronoi diagrams — structures we will
encounter in Chapter 6. Moreover, projection polyhedra can be found
efficiently [89].
The topic of lifting (not necessarily convex) cell complexes is briefly
addressed in Subsection 6.3.3. For equilibrium states in nonconvex
polygonal planar partitions, the Maxwell–Cremona theorem gives an
elegant answer; negative tensions of edges (stresses) will now occur in the
characterization. This theorem has surprising applications different from
its direct use in terrain recognition and scene analysis — for instance to
4.2. Delaunay optimization properties
41
the long-standing open problem of untangling simple polygons in the plane,
which has been solved in Connelli et al. [231].
4.2. Delaunay optimization properties
The Delaunay triangulation, DT(S), of a set S of n sites in the plane
possesses a host of nice and useful properties many of which are well known
and understood nowadays. As being the geometric dual of the Voronoi
diagram V (S), DT(S) comprises the proximity information inherent to S
in a compact manner. Apart from the present section, various properties of
DT(S) and their applications are described in Chapter 8 and, in particular,
in Section 8.2. Here we look at DT(S) as a triangulation per se and
concentrate on parameters which are optimized by DT(S) over all possible
triangulations of the point set S.
Recall that a triangulation (or triangular network ), T , of S is a maximal
set of non-crossing line segments spanned by the sites in S. In other words,
T defines a partition of the convex hull of S into triangles whose vertex
set is exactly S. The number of different such partitions is large (in fact
exponential, Ω(2.33n )) for every n-point set S in general position [43], which
makes the optimality results to be presented even more valuable.
Let us call T locally Delaunay if, for each of its convex quadrilaterals Q,
the corresponding two (adjacent) triangles have circumcircles empty of
vertices of Q. Clearly, DT(S) is locally Delaunay because the circumcircles
for its triangles are totally empty of sites; see Theorem 2.1. Interestingly,
the local property also implies the global one.
Theorem 4.2. A triangulation of S is locally Delaunay if and only if it
equals the Delaunay triangulation, DT(S).
Proof. Let T be a triangulation of S and assume that T is locally
Delaunay. We show that, for each triangle ∆ of T , its circumcircle C(∆) is
empty of sites in S.
Assuming the contrary, let s be inside C(∆) for some s ∈ S and some
∆ in T . Observe s ∈
/ ∆ and let e be the edge of ∆ closest to s. Suppose,
without loss of generality, that (∆, e, s) maximizes the angle spanned by e
at s, for all such triples (triangle, edge, site).
See Figure 4.6. Because of s, edge e cannot be an edge of the convex
hull of S. Let triangle ∆ be adjacent to ∆ at e, and let s be the third
vertex of ∆ . As T is locally Delaunay, s lies outside C(∆), hence s = s .
Now, observe that s is enclosed by C(∆ ), and let e be the edge of ∆
closest to s. The angle at s for (∆ , e , s) is larger than that for (∆, e, s),
which gives a contradiction.
Chapter 4. Advanced Properties
42
s
∆
∆’
e’
e
s’
Figure 4.6.
Illustration of the proof of Theorem 4.2.
An edge flip in a triangulation T of S is the exchange of the two
diagonals in one of T ’s convex quadrilaterals; see Section 3.2. Call an
edge flip good if, after the flip, the triangulation inside the quadrilateral
(consisting of only 2 triangles) is locally Delaunay. Repeated exchange
of diagonals in the same quadrilateral always produces an alternating
sequence of good and not good flips. Theorem 4.2 now can be used to
prove that DT(S) optimizes various quality measures, by showing that
each good flip increases quality. Any sequence of good flips then terminates
at the global optimum, the Delaunay triangulation. The length of such
a sequence is bounded by n2 , as an edge that gets flipped away cannot
reappear; see the argument on flip distances in planar triangulations given
in Subsection 6.3.3.
One of the most prominent quality measures concerns the angles
occurring in a triangulation. Recall that the number of edges (and thus
of triangles) does not depend on the way of triangulating S, and let t be
the number of triangles for S. The angularity of a triangulation is defined
to be the sorted list of angles (α1 , . . . , α3t ) of its triangles, in ascending
order. A triangulation is called equiangular if it possesses lexicographically
largest angularity among all possible triangulations for S.
As a matter of fact, every good flip increases angularity. Figure 4.7 gives
evidence for this fact. Lawson [486] called a triangulation locally equiangular
if no flip can increase angularity. Locally equiangular thus is equivalent
to locally Delaunay. Sibson [652] and Lee [492] first proved Theorem 4.2,
showing that locally equiangular triangulations are Delaunay and hence
are unique. Edelsbrunner [300] observed that DT(S) is equiangular (in the
global sense) as the global property implies the local one.
4.2. Delaunay optimization properties
Figure 4.7.
43
Equiangularity and the empty circle property.
In case of cocircularities among the sites, DT(S) is not a full
triangulation; see Chapter 2. Mount and Saalfeld [550] showed that DT(S)
can be completed by retaining local equiangularity, in O(n log n) time.
Theorem 4.3. Let S be a finite set of sites in the plane. A triangulation
of S is equiangular only if it is a completion of DT(S).
The equiangular triangulation obviously maximizes the minimum angle
that occurs in all the triangles. This property is desirable for mesh
generation applications in terrain modeling or for the finite element method,
as was first observed in Lawson [486] and McLain [528]. By Theorem 4.3,
such triangulations can be computed in O(n log n) time with the Delaunay
triangulation algorithms in Chapter 3.
More recently, it has been observed that several other parameters
are optimized by DT(S). All the properties listed below can be proved
by observing that every good edge flip locally optimizes the respective
parameter.
Consider the circumcircle for each triangle in a triangulation, and
measure coarseness by the largest such circle that arises; see Figure 4.7
again. As a matter of fact, DT(S) minimizes coarseness among all
possible triangulations for S; see D’Azevedo and Simpson [240]. We may
define coarseness also by taking smallest enclosing circles rather than
circumcircles. (Note that the smallest enclosing circle differs from the
circumcircle iff the triangle is obtuse.) D’Azevedo and Simpson proved that
DT(S) minimizes coarseness in this sense, and Rajan [598] showed that
this property of Delaunay triangulations — unlike others — generalizes to
higher dimensions.
44
Chapter 4. Advanced Properties
Similarly, fatness of a triangulation may be defined as the sum of
inradii, that is, radius of the largest circle inscribed to each triangle.
Lambert [480] showed that DT(S) maximizes fatness, or equivalently, the
mean inradius.
Given an individual function value (height) h(p) for each site p ∈ S,
every triangulation T of S defines a triangular surface in 3-space. The
roughness of such a surface may be measured by
|∆| · (α2 + β 2 ),
∆∈T
where |∆| is the area of ∆, and α and β denote the slopes that the
corresponding triangle in 3-space forms with the x-axis and the y-axis,
respectively. In other words, roughness is the integral of the squared
gradients. As has been shown by Rippa [605], roughness is minimum for
the surface obtained from DT(S), for any fixed heights h(p). For a simpler
proof, see Powar [594]. See also Musin [558] for various functionals on
triangular surfaces which are optimized by DT(S).
Let us mention that, in addition, DT(S) provides a means for smoothing
the corresponding triangular surface. As was shown in Sibson [653], each
point x within the convex hull of S can be expressed as the weighted centroid
of its Delaunay neighbors p in DT(S ∪{x}). Weights wp (x) can be computed
from area properties of the corresponding diagram regions, and as functions
of x, are continuously differentiable; see also Farin [336]. The corresponding
interpolant to the spatial points (p, h(p)) is called the natural neighbor
interpolant, and is given by
wp (x)h(p).
φ(x) =
p∈S
This useful property of DT(S) is shown to generalize to regular
triangulations (duals of power diagrams for S, see Section 6.2), and to
higher-order Voronoi diagrams (Section 6.5) in Aurenhammer [91]. For
more material on natural neighbor interpolants and their applications, see,
e.g., Boissonnat and Cazals [140].
On the negative side, DT(S) in general fails to fulfill optimization
criteria similar to those mentioned above, such as minimizing the maximum
angle, or minimizing the longest edge. Edelsbrunner et al. [315, 317] give
near-quadratic time algorithms for computing triangulations optimal in
that sense. DT(S) is not even locally minimal, in the sense that it does not
always include the shorter diagonal for each of its convex quadrilaterals.
Kirkpatrick [455] proved that DT(S) may differ arbitrarily strongly
from a minimum-weight triangulation, which is defined to have minimum
4.2. Delaunay optimization properties
45
total edge length. Computing a minimum-weight triangulation is an
important and interesting problem, whose complexity remained unknown
for a long time; see Garey and Johnson [352]. Recently, Mulzer and
Rote [557] showed its NP-hardness. (For the class of so-called NP-hard
problems, it is very unlikely that polynomial-time algorithms exist, i.e.,
algorithms with a running time of O(nk ), with k being a constant.) Subsets
of edges of DT(S) which always have to belong to a minimum-weight
triangulation are exhibited in Subsection 8.2.3.
On the other hand, the widely used greedy triangulation, which is
obtained by inserting non-crossing edges in increasing length order, can
be constructed from DT(S) in O(n) time, by a result in Levcopoulos and
Krznaric [504].
Finally, let us mention that the Delaunay triangulation avoids an
undesirable property that might be shared by other triangulations. Fix
a point v in the plane, called the viewpoint. For two triangles ∆ and ∆ in
a given triangulation, write ∆ < ∆ if ∆ fully or partially hides ∆ as seen
from v. This defines a relation, called the in-front/behind relation, on the
triangles. De Floriani et al. [245] observed that this relation is acyclic if the
triangulation is Delaunay, no matter where the viewpoint is chosen. That is,
the relation gives a partial order, also called a shelling order (with center v)
of the triangles, see e.g. Brugesser and Mani [171] and Subsection 6.5.1.
An example of a triangulation which is cyclic in spite of being minimumweight can be found in Aichholzer et al. [32].
Edelsbrunner [301] generalized the result in [245] to regular simplicial
complexes in d dimensions, a class that includes higher-dimensional
Delaunay triangulations as a special case; see Section 6.3. An application
stems from a popular algorithm in computer graphics that eliminates
hidden objects by first partially ordering the objects according to the infront/behind relation and then displaying them from back to front, thereby
overpainting invisible parts. In particular, this algorithm will work well for
α-shapes (Subsection 8.2.2) and their weighted variants (Section 6.6). A
rapid algorithm for displaying 3D Delaunay triangulations, or parts thereof,
on a standard raster display is described in Karasick et al. [445]; cf. also
Section 11.1 for pixel Voronoi diagrams.
For early structural discussions of planar triangulations, the reader is
referred to the theses by Tan [675] and Lambert [481], respectively. A survey
on optimization properties of triangulations is given in Aurenhammer and
Xu [106].
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Chapter 5
GENERALIZED SITES
In order to meet practical needs, the concept of Voronoi diagram has
been modified and generalized in many ways, for example by changing the
underlying space, the distance function used, or the shape of the sites.
Chapters 5 to 7 give a systematic treatment of generalized Voronoi diagrams.
It is commonly agreed that most geometric scenarios can be modeled
with sufficient accuracy by polygonal objects. Two typical and prominent
examples are the description of the workspace of a robot moving in the
plane, and the geometric information contained in a geographical map.
In both applications, robot motion planning and geographical information
systems, the availability of proximity information for the scenario is crucial.
This is among the reasons why considerable attention has been paid to the
study of Voronoi diagrams for polygonal objects.
Still, in some applications the scenario can be modeled more
appropriately when curved objects, for instance, circular arcs are also
allowed. Many Voronoi diagram algorithms working for line segments can
be modified to work for curved objects as well.
5.1. Line segment Voronoi diagram
Let G be a planar straight-line graph on n points in the plane, that is, a set of
non-crossing line segments spanned by these points. For instance, G might
be a tree (a connected graph without edge cycles), or a collection of disjoint
line segments or polygons (edge cycles), or a complete triangulation of the
points (a maximal non-crossing set of edges). The number of segments of
G is maximal, at most 3n − 6, in the last case. We will discuss several types
of diagrams for planar straight-line graphs in the present and following
sections.
The classical type is the (closest site) Voronoi diagram, V (G), of G. It
consists of all points in the plane which have more than one closest segment
47
48
Chapter 5. Generalized Sites
Figure 5.1.
Line segment Voronoi diagram.
in G. V (G) is known under different names in different areas, for example,
as the line Voronoi diagram or skeleton of G, or as the medial axis or the
symmetric axis when G is a simple polygon.
(A polygon is called simple if it is homeomorphic to a disk. Simple
polygons have connected boundaries; in particular, they contain no ‘holes’.)
Applications in such diverse areas as biology, geography, pattern
recognition, computer graphics, and motion planning exist; see e.g.
Kirkpatrick [454] and Lee [494] for early references, and the recent book
by Siddiqi and Pizer [654].
See Figure 5.1. V (G) is formed by straight-line edges and parabolically
curved edges, both shown as dashed lines. Straight-line edges are part of
either the perpendicular bisector of two segment endpoints, or of the angular
bisector of two segments. Curved edges consist of points equidistant from a
segment endpoint and a segment’s interior. There are two types of vertices,
namely of type 2 having degree two, and of type 3 having degree three
(provided that the point set which spans G is in general position). Both are
equidistant from a triple of objects (segment or segment endpoint), but for
type-2 vertices the triple contains a segment along with one of its endpoints.
5.1. Line segment Voronoi diagram
49
Together with G’s segments, the edges of V (G) partition the plane
into regions. These can be refined by introducing certain normals through
segment endpoints (shown dotted in Figure 5.1), in order to delineate faces
each of which is closest to a particular segment or segment endpoint. Two
such normals start at each segment endpoint where G forms a reflex angle,
and also at each terminal of G which is an endpoint belonging to only one
segment in G. A normal either ends at a type-2 vertex of V (G) or extends
to infinity.
Observe that each face f is visible from its defining component c in G;
that is, for each point x ∈ f there is a point y ∈ c such that their straightline connection xy stays entirely within f . (If c is a segment endpoint then
y = c, and if c is a segment then take for y the point on c closest to x.)
As a consequence, the union of the three faces for a segment of G plus its
two endpoints is a simply connected set, i.e., it is connected and contains
no holes.
It is well known that the number of faces, edges, and vertices of V (G) is
linear in n, the number of segment endpoints for G. The number of vertices
is shown to be at most 4n − 3 in Lee and Drysdale [496]. An exact bound,
which also counts the ‘infinite’ vertices at unbounded edges and segment
normals, is given below.
Lemma 5.1. Let G be a planar straight-line graph on n points in the plane,
and let G realize t terminals and r reflex angles. The number of (finite and
infinite) vertices of V (G) is exactly 2n + t + r − 2.
Proof. Suppose first that G consists of e disjoint segments (that do not
touch at their endpoints). Then there are e regions, and each type-3 vertex
belongs to three of them. By Euler’s formula for planar graphs, there are
exactly 2e − 2 such vertices, if we also count those at infinity. To count the
number of type-2 vertices, observe that each segment endpoint is a terminal
and gives rise to two segment normals each of which, in turn, yields one
(finite or infinite) vertex of type 2. Hence there are 4e such vertices, and
6e − 2 vertices in total.
Now let G be a general planar straight-line graph with e segments. We
simulate G by disjoint segments, by shortening each segment slightly such
that the segment endpoints are in general position. Then we subtract from
6e − 2 the number of vertices which have been generated by this simulation.
Consider an endpoint p that is incident to d ≥ 2 segments of G.
Obviously, p gives rise to d copies in the simulation.
The Voronoi diagram of these copies has d − 2 finite vertices, which are
new vertices of type 3. As the sum of the degrees d ≥ 2 in G is 2e − t, we
get 2e − t − 2(n − t) new vertices in this way.
50
Chapter 5. Generalized Sites
Each convex angle at p gives rise to two new normals emanating at the
respective copies of p, and thus to two (finite) type-2 vertices. A possible
reflex angle at p gives rise to one (finite or infinite) type-3 vertex, on the
perpendicular bisector of the corresponding copies of p. There are r reflex
angles in G, and thus 2e − t − r convex angles. This gives r + 2(2e − t − r)
new vertices in addition.
The lemma follows by simple arithmetic.
Surprisingly, the number of edges of G does not influence the bound
in Lemma 5.1. The maximum number of vertices, 3n − 2, is achieved, for
example, if G is a set of disjoint segments (t = n and r = 0), or if G is
a simple polygon P (t = 0 and r = n). In the latter case, the majority of
applications concerns the part of V (P ) interior to P . This part is commonly
called the medial axis of P . The medial axis of an n-gon with r reflex interior
angles has a tree-like structure and realizes exactly n + r − 2 vertices and
2(n + r) − 3 edges, a bound first mentioned in Lee [494].
Several algorithms for computing V (G), for general or restricted
planar straight-line graphs G, have been proposed and tested for practical
efficiency. V (G) can be computed in O(n log n) time and O(n) space by
divide & conquer algorithms (Kirkpatrick [454], Lee [494], and Yap [712]),
by using the plane sweep technique (Fortune [344]), and by randomized
incremental insertion (Boissonnat et al. [142] and Klein et al. [466]); cf.
Chapter 3 on basic algorithms.
Burnikel et al. [173] give an early overview of existing methods for
computing V (G). They also discuss implementation details of an algorithm
in Sugihara et al. [669] that first inserts all segment endpoints, and then
all the segments, of G in random order. An enhanced fast implementation
of this algorithm is given in Held [399]. A method of comparable simplicity
and practical efficiency (though with a worst-case running time of Θ(n2 )) is
proposed in Gold et al. [366, 367]. They first construct a Voronoi diagram
for point sites by selecting one endpoint for each segment, and then
maintain the diagram while expanding the endpoints, one by one, to their
corresponding segments. During an expansion, the resulting topological
updates in the diagram can be carried out efficiently.
In fact, Voronoi diagrams for moving sites, which are sometimes also
called kinetic Voronoi diagrams in the literature, are well-studied concepts;
they will be discussed in Section 9.1.
Recently, a practical and efficient divide & conquer algorithm, which
works for general curved sites, and constructs the diagram V (G) as a special
straight-line case, has been devised in Aichholzer et al. [25]. We will detail
this method in Section 5.5.
5.1. Line segment Voronoi diagram
51
If G is a connected graph then V (G) can be computed in randomized
time O(n log∗ n); see Devillers [261]. (Here log∗ n denotes the iterated
logarithm, that is, the number of times the (dual) logarithm has to be
applied to n in order to obtain a number smaller than 1.) Moreover, O(n)
time randomized, and deterministic, algorithms for the medial axis of a
simple polygon have been designed by Klein and Lingas [465] and Chin
et al. [218], settling open questions of long standing. The case of a convex
polygon is considerably easier; see Section 5.2.
An efficient O(n log2 n) work parallel algorithm for computing V (G)
is given in Goodrich et al. [373]. This is improved to O(log n)
parallel (randomized) time using O(n) processors in Rajesekaran and
Ramaswami [599]. The latter result also implies an optimal parallel
construction method for the Voronoi diagram for point sites.
The line segment Voronoi diagram V (G) is a planar graph embedded
in the plane, and as such has a well-defined combinatorial dual graph.
However, this dual reflects the adjacencies between the regions of V (G) in
a way not always desirable in applications. More specifically, the dual is a
‘triangulation’ of the segments in G that may contain multiple edges, due to
multiple adjacencies between Voronoi regions. This shortcoming is remedied
in Chew and Kedem [216], who define a so-called segment Delaunay
triangulation DT(G) for G, by placing certain additional points on G’s
segments. Based on this idea, Brévilliers et al. [165] define general segment
triangulations, and show that local optimality that characterizes the
classical Delaunay triangulation (Section 4.2) extends to these structures.
Moreover, they prove in [166] that DT(G), and thus, also the primal
structure, V (G), can be constructed by improving flips in segment
triangulations.
Along with the study of closest-site Voronoi diagrams go their farthestsite counterparts. In that model, each site s gets allotted the region of all
points for which s is the farthest site (rather than the closest) in the input
set. See Section 6.5 for the case of point sites, where the diagram has a
simple tree structure — just the sites on the convex hull of the input point
set contribute nonempty regions.
For line segments as sites, their farthest-site Voronoi diagram is
investigated in Aurenhammer et al. [95]. The properties of this diagram
deviate from the obvious. For example, regions may be disconnected, and
region emptiness cannot be characterized by convex hull properties; see
Figure 5.2. Moreover, and unlike the closest line segment case, the number
of edges and vertices of the diagram remains O(n) in the worst case when
the segments in the input set G do not form a planar graph (i.e., define
crossings).
Chapter 5. Generalized Sites
52
3
4
s6
5
s1
s4
s5
s2
s3
1
6
4
Figure 5.2. Farthest-segment Voronoi diagram for six sites. Encirculated numbers
indicate affiliation of regions to segments (from [95]).
The diagram can be computed in time O(n log n), by applying a
generalized (and primal) version of the ear-clipping algorithm for the
farthest-site Delaunay triangulation in Subsection 6.5.1.
Basically, a convex projection surface in 3-space is constructed, by
using, for each point x in the plane, its distance to the farthest segment
in G as a third coordinate for x — an algorithmic idea that also works for
the medial axis or the straight skeleton of a convex polygon; cf. Sections 5.2
and 5.3. For nonconvex simple polygons such an algorithm would however
fail, due to the nonconvexity of the resulting projection surface.
The farthest-segment Voronoi diagram is still a tree; all its connected
faces are unbounded. The upper bound of 8n + 4 in [95] on their number
has been improved recently to 6n − 6 in Papadopoulou and Dey [580].
The farthest-site Voronoi diagram for simple polygons as sites is studied
in Cheong et al. [208]. Its size is still O(n), though its structure is more
complicated, leading to an O(n log3 n) time algorithm.
Line segment Voronoi diagrams of order k (cf. Section 6.5) are
another natural generalization, covering the closest line segment Voronoi
diagram (k = 1) and the farthest-segment Voronoi diagram (k = |G| − 1).
Surprisingly, they did not receive much attention for general k until
5.2. Convex polygons
53
recently, when Papadopoulou and Zavershynskyi [584] showed that their
combinatorial complexity remains in O(k(n − k)), as in the point site case.
In dimensions more than two, the known results on Voronoi
diagrams for generalized sites are sparse. Exceptions are power diagrams
(Section 6.2), Voronoi diagrams for spheres, and the medial axis in three
dimensions (both Section 6.6).
5.2. Convex polygons
Voronoi diagrams for a single convex polygon have a particularly
simple structure. Tailor-made algorithms for their construction have been
designed.
Let C be a convex n-gon in the plane. The part of the Voronoi diagram
of C’s sides which lies inside C is called the medial axis, M (C), of C. It
is a tree whose edges, by convexity of C, are pieces of angular bisectors of
the sides. In fact, M (C) coincides with the straight skeleton SK(C) of C
(discussed in Section 5.3) in this case. See Figure 5.3 for an illustration.
M (C) realizes exactly n faces, n − 2 vertices, and 2n − 3 edges.
There is a simple randomized incremental algorithm by Chew [210]
that computes M (C) in O(n) time. The algorithm first removes, in random
order, the halfplanes whose intersection is C. Removing a halfplane means
removing the corresponding side e of C, and extending the two sides e
and e adjacent to e so that they become adjacent in the new convex
polygon. This can be done in constant time per side. (If the extensions of e
and e do not intersect then the obtained polygon is unbounded but still
convex; we omit the easy modifications necessary for that case.)
Figure 5.3.
Medial axis of a convex polygon.
54
Chapter 5. Generalized Sites
The adjacency history of C is stored. That is, for each removed side e,
one of its formerly adjacent sides is recorded. In a second stage, the sides
are put back in reverse (still randomized) order, and the medial axis is
maintained during these insertions.
Let us focus on the insertion of the ith side, ei . We have to integrate
the face, f (ei ), of ei into the medial axis of the i − 1 sides that have been
inserted before ei . From the adjacency history, we already know a side e
of the current polygon that will be adjacent to ei after its insertion. Hence
we know that the angular bisector of ei and e will contribute an edge
to f (ei ).
Having a starting edge available in O(1) time, the face f (ei ) now can
be constructed in time proportional to its size. We construct f (ei ) edge by
edge, by simply tracing and deleting parts of the old medial axis interior
to f (ei ). As the medial axis of an i-gon has 2i − 3 edges, and each edge
belongs to two faces, the expected number of edges of a randomly chosen
face is less than 4. Thus f (ei ) can be constructed in constant expected time,
which gives an O(n) randomized algorithm for computing M (C).
The same technique also applies to the Voronoi diagram for the vertices
of a convex n-gon C, that is, to the Voronoi diagram of a set S of n point
sites in convex position. By Lemma 2.2, all regions in V (S) are unbounded,
such that the edges of V (S) have to form a tree. Hence V (S) has the same
number of edges and (finite) vertices as the medial axis of C.
For each p ∈ S, its region VR(p, S) shares an unbounded edge with the
regions VR(p , S) and VR(p , S), respectively, where p and p are
the neighbors of p on the boundary of the convex hull of S (which is
the polygon C). An adjacency history can be computed in O(n) time, by
removing the sites in random order and maintaining their convex hull. For
each site that is re-inserted, the expected number of edges is less than 4, as
before. So an O(n) randomized construction algorithm is obtained.
The diagrams V (S) and M (C) can also be computed in deterministic
linear time; see Aggarwal et al. [17]. The details of this algorithm are much
more involved, however.
5.3. Straight skeletons
In comparison to the Voronoi diagram for point sites, which is composed
of straight edges, the occurrence of curved edges in the line segment
Voronoi diagram V (G) is a disadvantage in the computer representation
and construction, and sometimes also in the application, of V (G).
There have been several attempts to linearize and simplify V (G),
mainly for the sake of efficient point location and motion planning; see
5.3. Straight skeletons
55
Canny and Donald [181], Kao and Mount [439], de Berg et al. [242], and
McAllister et al. [527]. The compact Voronoi diagram in [527] is particularly
suited to these applications. It is defined for the case where G is a set of
k disjoint convex polygons. Its size is only O(k), rather than O(n), and it
can be computed in time O(k log n); see Subsection 8.1.1 for more details.
As a different alternative to V (G), we now describe the straight
skeleton, SK(G), of a planar straight-line graph G. This structure
is introduced, and discussed in more detail, in Aichholzer and
Aurenhammer [29]. SK(G) is composed of angular bisectors and thus does
not contain curved edges. In general, its size is even less than that of V (G).
Its use as a type of skeleton for G partially stems from the fact that
G can be reconstructed from SK(G) in a simple manner. This fact is
considered important, for example, in picture processing; see Pfaltz and
Rosenfeld [590], who observe this property for the medial axis of a simple
polygon. The recently introduced triangulation axis of a polygon P also
shares this feature. This skeletal axis is piecewise linear, too, and is defined
via some optimal triangulation of P . Its size is (sometimes drastically)
smaller than the size of the medial axis or the straight skeleton of P ; see
Aigner et al. [44].
The relevance of the straight skeleton SK(G) in shape and object
recognition is discussed, e.g., in Tanase and Veltkamp [677] and Demuth
et al . [256]. Applications in robotics and to origami design have been
described in Barequet et al. [113], and in Lang [483] and Demaine et al. [255],
respectively. A lifted 3D model of SK(G) turns out to be of use in
architecture and geographic information systems, as will be sketched later.
(Interestingly, and seemingly having faded into oblivion, utilizing straight
skeletons as a means for designing roof constructions dates back more
than a hundred years, as is documented in a monograph for engineers
by Peschka [589], and later in a textbook on descriptive geometry by
Müller [553].) Straight skeletons also appear implicitly in a certain Voronoi
diagram for time distances, the so-called city Voronoi diagram for point
sites under a rectilinear transportation network; see Aichholzer et al. [38].
We will give details on the city Voronoi diagram in Section 7.6.
SK(G) is defined as the interference pattern of certain wavefronts
propagated from the segments and segment endpoints of G. Let F be a
connected component (called a figure) of G. Imagine F as being surrounded
by a belt of (infinitesimally small) width ε. For example, a single segment
s gives rise to a rectangle of length |s| + 2ε and width 2ε, and a simple
polygon P gives rise to two offset copies of P with minimum distance 2ε.
In general, if F partitions the plane into c connected domains then F gives
rise to c simple polygons, called wavefronts for F .
56
Chapter 5. Generalized Sites
The wavefronts arising from all the figures of G are now propagated
simultaneously, at the same speed, and in a self-parallel manner. Wavefront
vertices move on angular bisectors of wavefront edges which, in turn,
may increase or decrease in length during the propagation. This situation
continues as long as the wavefronts do not change combinatorially. Basically,
there are two types of changes, called events.
(1) Edge event : A wavefront edge collapses to length zero. (The wavefront
may vanish altogether, due to three simultaneous edge events.)
(2) Split event : A wavefront edge splits due to interference or selfinterference. In the former case, two wavefronts merge into one, whereas
in the latter case a wavefront splits into two.
After either type of event, we are left with a new set of wavefronts
which are propagated recursively.
The edges of SK(G) now are the pieces of angular bisectors traced
out by wavefront vertices. Each vertex of SK(G) corresponds to some point
where an edge event or a split event takes place. SK(G) is a unique structure
defining a polygonal partition of the plane; see Figure 5.4.
During the propagation, each wavefront edge e sweeps across a certain
area which we call the face of e. Each segment of G gives rise to two
wavefront edges and thus to two faces, one on each side of the segment.
Figure 5.4.
Straight skeleton for three figures.
5.3. Straight skeletons
57
Each terminal of G (endpoint of degree one) gives rise to one face. Faces
can be shown to be monotone polygons, i.e., each intersection of a face with
lines having some fixed direction is connected. In our case, this direction is
normal to the generating wavefront edge. As a consequence, faces are simply
connected sets. This implies a total of 2m + t = O(n) faces, if G realizes
m edges and t terminals. There is also an exact bound on the number of
vertices of SK(G).
Lemma 5.2. Let G be a planar straight-line graph on n points, and t of
which are terminals. The number of (finite and infinite) vertices of SK(G)
is exactly 2n + t − 2.
From Lemma 5.1 it is apparent that SK(G) has r vertices less than V (G),
if G defines r reflex angles. In particular, if G is a simple polygon with r
reflex interior angles, then the part of SK(G) interior to G is a tree with
only n − 2 vertices, whereas the medial axis of G has n + r − 2 vertices.
By construction, SK(G) encodes all self-parallel inward and outward
offsets for G, also called mitered offsets in the polygon case; see e.g.
Devadoss and O’Rourke [260]. However, this offsetting process is different
from the classical offsetting process, which is based on a wavefront model
(or growth model ) for the Voronoi diagram or the medial axis V (G) of G; cf.
the end of Section 5.5, and the ‘expanding waves’ view in Chapter 2. The
propagation speed of all points on the wavefront is the same in that model,
whereas, in the model for SK(G), the speed of each wavefront vertex, u, is
controlled by the interior angle, β(u), between its incident wavefront edges.
The sharper the angle, the faster is the movement of the vertex. More
precisely, the speed s(u) of vertex u is given by the formula
s(u) = 1/ sin
β(u)
2
and tends to infinity if the angle β(u) approaches 0. This behavior makes
SK(G) completely different from the Voronoi diagram of G. It can be
shown that, without prior knowledge of its structure, SK(G) cannot be
defined by means of distances from G, by lack of its non-procedural
definition; see [28]. Also, SK(G) does not fit into the framework of abstract
Voronoi diagrams described in Section 7.5: The bisector of two segments
of G would be the interference pattern of the rectangular wavefronts they
send out, but these curves do not fulfill condition (ii) in Definition 7.7.
As a consequence, the well-developed machinery for constructing Voronoi
diagrams (see Chapter 3) does not apply to SK(G).
An algorithm that simulates the wavefront propagation is given in [29].
It maintains a triangulation, T , of the wavefront vertices in ‘free space’ —
58
Chapter 5. Generalized Sites
the part of the plane not yet swept over by any wavefront. Collision of
offsetting figures can be predicted by the collapse of triangles in T . Though
the known upper bounds are much higher, T typically undergoes only O(n)
updates (of several kinds, including collapses). A runtime of O(n log n)
is then achieved, so the method is simple and practically efficient. We
conjecture that the techniques for kinetic collision detection in Kirkpatrick
et al. [459], which are based on a so-called pseudo-triangulation (Section 6.3)
of the free space, are capable of theoretically speeding up this approach.
More sophisticated though more involved construction methods
running in subquadratic time are given in Eppstein and Erickson [326],
Cheng and Vigneron [205], and recently in Vigneron and Yan [691] who
achieve a running time of O(n4/3+ε ). These works also describe additional
structural properties of the straight skeleton. The so-called motorcycle graph
is introduced and utilized, which captures an important aspect of straight
skeletons. This planar graph resolves the interplay of the reflex vertices’
trajectories, and thus is of substantial help in predicting the (by nature
non-local) split events. The paper by Vyatkina [695] shows that SK(G)
can be decomposed into (pruned) medial axes of certain convex polygons.
Recently, practical implementations of straight skeleton algorithms have
been developed in Huber and Held [410] and in Palfrader et al. [577].
Whether SK(G) can be computed in O(n log n) time, and possibly even
faster inside a simple polygon, is still an open problem. In fact, only the
trivial lower bound, Ω(n), is known in the polygon case.
In view of the notoriously difficult task of computing the straight
skeleton SK(G) efficiently, simpler instances for G have been studied as
well. Clearly, if G is a convex polygon, then SK(G) is just the medial axis
of G, which can be constructed in O(n) time (Section 5.2).
More interesting is the case of a monotone polygon, which is solvable
in O(n log n) time; see Das et al. [239]. Basically, the polygon’s boundary
is decomposed into two monotone chains of edges. The straight skeleton
of each chain can be constructed without the trouble of split events, and
the two partial skeletons merged in a simple way, similar as in the Voronoi
diagram case; see Section 3.3.
SK(G) has a three-dimensional interpretation, obtained by defining the
height of a point x in the plane as the unique time when x is reached by
a wavefront. In this way, SK(G) lifts up to a polygonal surface ΣG , where
points on G have height zero. In a geographical application (e.g., terrain
modeling), G may delineate rivers, lakes, and coasts, and ΣG represents
a corresponding terrain with fixed slope; see Figures 5.5 and 5.6 (taken
from [29]). ΣG has the nice property that every raindrop that hits a terrain
facet f drains off to the segment or terminal of G defining f ; see [28]. This
5.3. Straight skeletons
Figure 5.5.
Coastline and rivers.
Figure 5.6.
59
Reconstructed terrain.
may have applications in the study of rain water fall and its impact on the
flooding caused by rivers in a given geographic area, where currently the
Voronoi diagram of G is used; see, for example, Barett [116].
When restricted to the interior of a simple polygon P , the surface ΣP
can be used as a canonical construction of a roof of given slope above P .
For rectilinear (and axis-aligned) polygons P , the medial axis of P in the
L∞ -metric will do the job. SK(P ) coincides with this medial axis for
such polygons, and thus generalizes this roof construction technique to
general shapes of P . Applications arise, e.g., in the automated generation of
building models; see Brenner [164], Kelly and Wonka [451], and references
therein.
Sometimes, the volatile behavior of the straight skeleton SK(P ), which
stems from the high propagation speed of sharp reflex vertices of the
polygon P , is considered a disadvantage. A possible way out is to ‘cut
off’ such vertices in a controlled way, an idea sketched in [29] and applied
systematically in Tanase and Veltkamp [676]. The resulting (appropriately
pruned) straight skeleton is called the linear axis of the polygon P in the
latter paper, and is shown to approximate the medial axis of P in a certain
sense. See also Bookstein [153] for an early study of the straight skeleton
for polygons with large interior angles, who termed it the line skeleton.
The concept of straight skeleton SK(G) of a planar straight line graph G
can be modified by tuning the propagation speed of the individual wavefront
edges. That is, edges can be given a weight that expresses their velocity. For
the 3D model of SK(G) described above, this means putting prescribed facet
slopes for the corresponding surface. The maximal size of the skeleton, and
its procedural definition, remain unaffected. (The exceptional case where
60
Chapter 5. Generalized Sites
two parallel edges of different weights become adjacent after an edge event
can be treated consistently, by considering the created wavefront vertex as
a convex vertex.) The monotonicity of the skeleton faces, however, and if G
is disconnected even their simple connectedness, may be lost.
In the particularly interesting case where G is the boundary of a
convex polygon P , this so-called weighted skeleton, SKW (P ), is studied in
Aurenhammer [94]. By adjusting the edge weights accordingly, SKW (P ) is
capable of partitioning a given n-gon P into n convex parts, each part being
based on a single edge of P and covering a predefined ‘share’ of P . The share
may relate, for example, to the spanned area, to the number of contained
points from a given point set, or to the total edge length covered in a given
set of curves. (The partitioning results for power diagrams presented in
Section 6.4 are related but not equivalent.) Possible applications of such
fixed-share decompositions include priority-based or fair facility allocation,
which may concern real estate or access to power lines, aquifers, or oil
wells.
Area decomposition may be done recursively, yielding a new instance
of a structure called Voronoi tree map, of use for visualizing attributed
hierarchical data; see Andrews et al. [62], Granitzer et al. [377], and Balzer
and Deussen [111].
Straight skeletons can be generalized to higher dimensions. For
example, in 3D the input could be a nonconvex and boundary-connected
polyhedron P with n vertices. (For convex polyhedra, the straight skeleton
coincides with the medial axis.) The shrinking process now involves
offsetting inwards the boundary facets of P in a self-parallel way and at
unit speed. Thereby, facets change their shape and may disconnect (as
do polygon edges in the 2D case), and the shrinking process continues with
each part. The volumes traced out by the facets are polyhedra again. Hence,
the structure retains its piecewise-linear shape and offers an alternative to
the medial axis of P whose geometric description is rather complex; see
Section 6.6.
However, and unlike the planar case, there is no unique (mitered)
offsetting process for P , in general. This was observed in Demaine et al. [254]
and Barequet et al. [115]. In fact, as there possibly exist several different
‘skeleton-like’ partitions of a nonconvex polygon by means of angular
bisectors [28], even such having super-linear size, there may be different
ways of locally offsetting a vertex v of P of high degree. In certain
cases, the planar weighted straight skeleton can be used to compute
a canonical offset structure [115]. In the general case (when the edges
incident to v positively span 3-space), a generalization of the straight
skeleton to the sphere will work; see Aurenhammer and Walzl [105].
5.3. Straight skeletons
61
r
v
c
Figure 5.7. Two local offset surfaces for a saddle point (right). The spherical skeleton
(left, dashed style) produces the lower solution. The short skeleton arc corresponds to a
convex edge c in the offset polytope. It connects the two vertices which the saddle point
is split into. In the upper solution, the offset edge r is reflex.
Figure 5.7 shows two possible offsets for a saddle point v of degree 4. Either
choice leads to a different though valid straight skeleton for the underlying
polyhedron P .
Concerning size, O(n2 α(n)2 ) is a lower bound on the combinatorial
complexity of a 3D skeleton [115]. (Here, α(n) denotes the inverse of the
rapidly growing Ackermann function; see e.g. [642]. We have α(n) ≤ 5 even
for ‘astronomically large’ arguments n, such that α(n) can be considered a
constant for all practical purposes.) A trivial upper bound is O(n4 ), because
each 4-tuple of facet planes for P can yield only one common point of
intersection during the entire offsetting process.
If P is an orthogonal polyhedron (all edges of P are parallel to the
coordinate axes) then the skeleton size reduces to O(n2 ), and an O(n2 log n)
construction algorithm exists [115]. A recent implementation using the
plane sweep technique is given in Martinez et al. [516]. The straight skeleton
is the L∞ -medial axis of P in this case, which also enables its efficient
computation in voxel representation. Yet, by its non-interpretability in
terms of distances in general [28], one cannot resort to powerful distance
transforms (like the EDT for the medial axis; see Section 11.1), which
makes the accurate computation of pixel straight skeletons an interesting
task already in 2D; see Demuth et al. [256].
An efficient algorithm for the straight skeleton of general nonconvex
3D polyhedra (in either representation) is still missing, as are non-trivial
upper bounds on its combinatorial complexity.
62
Chapter 5. Generalized Sites
5.4. Constrained Delaunay and relatives
In certain situations, unconstrained proximity among a set of sites is not
enough information to meet practical needs. There might be reasons for not
considering two sites as neighbors although they are geometrically close
to each other. For example, two cities that are geographically close but
separated by high mountains might be far from each other from the point
of view of a truck driver. The concepts described below have been designed
to model constrained proximity among a set of sites.
Let S be a set of n point sites in the plane, and let L be a set of noncrossing line segments spanned by S. Note that |L| ≤ 3n − 6. The segments
in L are viewed as obstacles: We define the bounded distance between two
points x and y in the plane as
b(x, y) =
d(x, y) if xy ∩ L = ∅,
∞
otherwise,
where d stands for the Euclidean distance. In the resulting bounded Voronoi
diagram V (S, L), regions of sites that are close but not visible from each
other are clipped by segments in L. Regions of sites being segment endpoints
are nonconvex near the corresponding segment; see Figure 5.8 (left).
The dual of V (S, L) is not a full triangulation of S, even if the segments
in L are included. However, V (S, L) can be modified to dualize into a
triangulation which includes L and, under this restriction, is as much
‘Delaunay’ as possible.
Figure 5.8. Bounded Voronoi diagram extended, and its dual, the constrained Delaunay
triangulation. The single constraining segment is drawn in bold style.
5.4. Constrained Delaunay and relatives
63
The modification is simple but nice. For each segment ∈ L, the regions
clipped by from the right are extended to the left of , as if only the
sites of these regions were present. The regions clipped by from the left
are extended similarly. See Figure 5.8 again, where these modifications are
shown in dotted lines. Of course, extended regions may overlap now, so
they fail to define a partition of the plane. If we dualize now by connecting
sites of regions that share an edge, a full triangulation that includes L is
obtained: the constrained Delaunay triangulation, CDT(S, L). It is clear
that the number of edges of CDT(S, L) is at most 3n − 6, and that, in
general, the number of edges of V (S, L) is even less. Hence both structures
have a linear size.
The original definition of CDT(S, L) in Lee and Lin [497] is based
on a modification of the empty-circle property: CDT(S, L) contains L
and, in addition, all edges between sites p, q ∈ S that have b(p, q) < ∞
and that lie on some circle enclosing only sites r ∈ S with at least one
of b(r, p), b(r, q) = ∞.
Algorithms for computing V (S, L) and CDT(S, L) have been proposed
in Lee and Lin [497], Chew [211], Wang and Schubert [697], Wang [696],
Seidel [629], and Kao and Mount [440]. The last two methods seem best
suited to implementation. For an application of CDT(S, L) to quality mesh
generation see Chew [213] and Shewchuk [647]; the latter paper carefully
discusses the implementation details.
We sketch the O(n log n) time plane sweep approach in [629]. If only
V (S, L) is required then the plane sweep algorithm described in Section 3.4
can be applied without much modification. If CDT(S, L) is desired then the
extensions of V (S, L) as described above are computed in addition. To this
end, an additional sweep is carried out for each segment ∈ L. The sweep
starts from the line through in both directions. It constructs, on the left
side of this line, the (usual) Voronoi diagram of the sites whose regions in
V (S, L) are clipped by from the right, and vice versa.
The special case where S and L are the sets of vertices, and sides,
of a simple polygon has received special attention, mainly because of
its applications to visibility problems in polygons. The bounded Voronoi
diagram V (S, L) is constructible in O(n) randomized time in this case; see
Klein and Lingas [463]. If the L1 -metric instead of the Euclidean metric
is used to measure distances, the same authors [465] give a deterministic
linear time algorithm. Both algorithms, as well as the linear time medial axis
algorithms in [464] and in [218], first decompose the polygon into smaller
parts called histograms. These are polygons whose vertices, when considered
in cyclic order, appear sorted in some direction.
64
Chapter 5. Generalized Sites
An alternative concept that forces a set L of line segments spanned by S
into DT(S) is the conforming Delaunay triangulation. For each segment
∈ L that does not appear in DT(S), new sites on are added such that becomes expressible as the union of Delaunay edges in DT(S ∪ C), where
C is the total set of added sites. For several site adding algorithms, |C|
depends on the size as well as on the geometry of L. See, e.g., the survey
article by Bern and Eppstein [126] and references therein. Edelsbrunner
and Tan [316] show that |C| = O(k 2 n) is always sufficient, and construct a
set of sites with this size in time O(k 2 n + n2 ), for k = |L|.
Conforming triangulations should not be confused with the concept of
compatible triangulations; see Aichholzer et al. [37]. These are triangulations
T1 and T2 of two planar point sets S1 and S2 , respectively, such that there is
a triangle-preserving bijection S1 → S2 . Whether a compatible triangulation
T2 always exists, given S1 , S2 , and T1 (under the obvious necessary size
and convex hull restrictions for the point set S2 ), is still an unsettled
question.
Things get remarkably different if the constraining line segments
(obstacles) in L do not have their endpoints included in the set S of
sites. In this setting, the obstacles will just block visibility, without exerting
proximity influence — a scenario useful in visibility, motion planning, and
guarding problems. The resulting visibility-constrained Voronoi diagram
is a piecewise linear structure of superlinear combinatorial complexity
in n = |S|. Its regions are disconnected in general, being the union of
polygons bounded by bisectors and visibility rays. Already for |L| = 2,
when visibility is constrained to a ‘window’ between two segments, and
the so-called peeper’s Voronoi diagram is obtained, the size may increase
to Θ(n2 ); see Aurenhammer and Stöckl [103]. This bound remains valid
if general visibility angles (of less than π) are attached to the sites, as in
Figure 5.9; see Fan et al. [335]. O(n2 ) and O(n2 log n) time algorithms,
respectively, exist for these scenarios.
For the general case of n sites among m line segment obstacles,
where the combinatorial complexity may be as large as Θ(n2 m2 ), Wang
and Tsin [698] gave an O(n4 + n2 m2 ) algorithm, worst-case optimal for
m = Ω(n).
A different, and geometrically even more complicated, type of
constrained Voronoi diagram is the geodesic Voronoi diagram, also called
the Voronoi diagram for obstacles. Here, the geodesic distance between a
site p and a point x in the plane is the length of the shortest obstacleavoiding path between p and x. It leads to a diagram with (path-)connected
regions and linear size; see Aronov [66]. Once the geodesic Voronoi diagram
is available, several proximity questions that respect the obstacles can be
5.4. Constrained Delaunay and relatives
65
1
3
3
1
1
4
3
4
4
3
1
2
3
2
1
1
2
1
2
2
2
4
3
3
4
φ
Figure 5.9. Voronoi diagram of four sites, constrained by angular visibility. The plane
is dissected by visibility rays into cells A(T ), exclusively seen by subsets T ⊆ S, and
partitioned by their
(classical) Voronoi diagram V (T ). The region of a site p can
S
be expressed as T p (A(T ) ∩ VR(p, T )), where VR(p, T ) is the region of p in V (T ).
The shaded areas above constitute the region of site 4. Cells not visible from S, like the
bottommost cell A(∅), belong to no site.
answered efficiently, like nearest neighbor queries or the construction of a
geodesic minimum spanning tree; cf. Chapter 8. The obstacles are usually
modeled by a set L of non-crossing line segments. If all their endpoints are
sites then the bounded Voronoi diagram is obtained.
Computing geodesic distances is not a constant-time operation,
which complicates matters. Bisectors may consist of various linear and
hyperbolic pieces. A subquadratic construction algorithm is given by
Mitchell [546], and Hershberger and Suri [404] speeded up his wavefront
propagation method (based on the continuous Dijkstra technique) to
O((n + m) log(n + m)) time and space, for n sites and m segments. The
storage is reduced to optimal, O(n + m), for the geodesic Voronoi diagram
inside a simple m-gon, in Papadopoulou and Lee [582].
Voronoi diagrams induced by geodesic distances (shortest paths) on
surfaces in 3-space are discussed in Section 7.1.
66
Chapter 5. Generalized Sites
There are many other ways of measuring distances in the plane in
the presence of line segments, or more generally, of polygonal objects or
straight-line graphs. Let us briefly mention two of them at this place. Let A
and B be two planar straight-line graphs. The Hausdorff distance between
A and B is defined as the maximum of the minimum distances from A to B,
and from B to A, respectively. An instance of the Voronoi diagram under
this distance (the so-called cluster Voronoi diagram) will be discussed in
Section 6.5.
The (continuous) Fréchet distance between A and B is based on
parametrizing A and B; for simplicity, assume that A and B are two
polygonal paths. As an intuitive interpretation, imagine a person walking
all along A and his/her dog walking all along B. Then the Fréchet distance
between the paths A and B corresponds to the minimum possible length of
a leash that enables both, the person and the dog, to complete their walk.
Computing Fréchet distances is quite elaborate. See Alt and Godau [51] for
an efficient algorithm, which is based on the parametric search paradigm
introduced in Megiddo [530].
Using a discrete version of this distance, Bereg et al. [125] recently
analyzed the combinatorial complexity of the respective Voronoi diagram
for a set of polygonal paths in two (and higher) dimensions.
5.5. Voronoi diagrams for curved objects
Some of the algorithms mentioned in Section 5.1 that compute the Voronoi
diagram for line segments will also work for curved objects.
The plane sweep algorithm in Fortune [344] elegantly handles arbitrary
sets of circles (i.e., the additively weighted Voronoi diagram, or Johnson–
Mehl model; see Section 7.4) without modification from the point site case.
Circular arcs, however, cannot be treated easily with that approach. Yap’s
divide & conquer algorithm [712] allows sets of disjoint segments of arbitrary
degree-two curves. A randomized incremental algorithm for general curved
objects is given by Alt et al. in [50]. They show that complicated curved
objects can be partitioned into ‘harmless’ ones by introducing new points.
All these algorithms achieve an optimal running time, O(n log n), but are
not easy to implement.
The topic of this section is a simple and practical algorithm for
computing the Voronoi diagram of a set of general (not necessarily
polygonal) sites, developed in Aichholzer et al. [25]. In fact, such diagrams
may have all kinds of artifacts. Their edge graph may be disconnected, and
their bisectors may be closed curves. In particular, the abstract Voronoi
5.5. Voronoi diagrams for curved objects
67
diagram setting in Klein [461] and Klein et al. [466] (see Section 7.5) does
not apply. Moreover, divide & conquer is usually involved when emphasis
is on the bottom-up phase (i.e. the merge step; cf. Section 3.3), even if the
sites are of relatively simple shape. This is due to the missing separability
condition for the sites, which would prevent the merge chain from cycling
and breaking into several components.
The idea is to put emphasis on the top-down phase, namely, the divide
step. The edge graph of the Voronoi diagram can be divided into a certain
tree that corresponds to the medial axis of a (generalized) planar domain,
as will be described in Subsection 5.5.1. (Usually, it is the other way
round: Voronoi diagram algorithms are applied to compute the medial axis;
cf. Section 6.6.) Division into base cases is then possible, which, in the
bottom-up phase, can be merged by trivial concatenation. The resulting
construction algorithm, in Subsection 5.5.2, is not bisector-based and merely
computes dual links between the sites — similar to Delaunay triangulation
methods. This guarantees computational simplicity and numerical stability.
No part of the Voronoi diagram, once constructed, has to be discarded
again.
5.5.1. Splitting the Voronoi edge graph
In the Voronoi diagram for general objects to be considered now, sites
are allowed to be two-dimensional objects (any topological disks), onedimensional objects (topological line segments), or simply points.
Let S denote the given set of sites. The distance of a point x to a
site s ∈ S is given by miny ∈ s d(x, y), where d denotes the Euclidean distance
function. As done e.g. in [50, 712], we define the Voronoi diagram, V (S),
of S via its edge graph, GS , which is the set of all points having more than
one closest point on the union of all sites. Our aim is to relate the graph GS
to the medial axis of a generalized planar domain. In this way, we will be
able to construct the Voronoi diagram V (S) by means of the medial axis
algorithm presented in Subsection 5.5.2.
An edge of GS containing points equidistant from two or more different
points on the same site s is called a self-edge for s. The regions of V (S) are
the maximal connected subsets of the complement (of the closure) of GS
in the plane. The differences to a bisector-based definition of the Voronoi
diagram should be noticed. Self-edges are ignored in such a definition
unless the sites are split into suitable pieces. Such pieces, however, share
boundaries — a fact that, if not treated with care, may give rise to
unpleasant phenomena like two-dimensional bisectors.
68
Chapter 5. Generalized Sites
To get rid of the unbounded components of the diagram, we include a
surrounding circle, Γ, into the set S of sites, in a way such that each vertex
of V (S\{Γ}) is also a vertex of V (S).
Removal of certain points from the edge graph GS of V (S) will now
break all its cycles and make it a tree. Finding such points is non-trivial,
in view of the possible presence of self-edges. For a site s = Γ, let p(s) be
a point on s with smallest y-coordinate, and denote with q(s) the closest
point on GS vertically below p(s). Then the geometric graph
TS = GS \{q(s) | s ∈ S\{Γ}}
can be shown to be a tree.
Our next goal is to interpret the tree TS as the medial axis of a
generalized planar domain. We first consider V (S) to be the medial axis
of a planar shape B, by taking the surrounding circle Γ as part of the
shape boundary, and considering each remaining site s ∈ S as a (possibly
degenerate) ‘hole’. That is, we define
B = B0 \{s ∈ S | s = Γ},
where B0 denotes the disk bounded by Γ. The medial axis M (B) is just the
(closure of the) edge graph GS of V (S). We now want to combinatorially
disconnect the shape B at appropriate positions, such that the medial axis
of the resulting domain corresponds to the tree decomposition TS of V (S)
above. Observe from Lemma 5.4 (in the next subsection, 5.5.2) that any
maximal inscribed disk can be used to split the medial axis of a shape
into two (or more) components which share a point at the disk’s center. In
order to adapt this fact to our situation, the notion of a so-called augmented
domain is introduced. Its definition is recursive, as follows.
An augmented domain is a set A together with a projection πA : A →
2
R . Initially, A is the original shape B, and the associated projection πB
is the identity. Now, consider a maximal inscribed disk D of an augmented
domain A, which touches the boundary ∂A of A at exactly two points u
and v. Denote by uv and vu the two circular arcs which the boundary of D
is split into. The new augmented shape, A , which is obtained from A by
splitting it with D, is defined as
A = A0 ∪ D1 ∪ D2 ,
where A0 = {(x, 0) | x ∈ A\D}, D1 = {(x, 1) | x ∈ D}, and D2 = {(x, 2) |
x ∈ D}. The associated projection is
πA : A → R2 ,
(x, i) → πA (x).
5.5. Voronoi diagrams for curved objects
69
We say that the line segment in A between points (x, i) and (y, j) is
contained in A if one of the following conditions is satisfied:
(1) i = j and the line segment xy avoids ∂D,
(2) {i, j} = {0, 1} and xy intersects the arc uv, or
(3) {i, j} = {0, 2} and xy intersects the arc vu.
For any two points (x, i) and (y, j) in A , their distance now can be
defined. It equals the distance of πA (x) and πA (y) in the plane, provided
the connecting line segment is contained in A , and is ∞, otherwise. An
(open) disk in A with center (m, i) and radius is the set of all points
in A whose distance to (m, i) is less than . Such a disk is said to be
inscribed in A if its projection into the plane is again an open disk.
Having specified inscribed disks for A , the boundary of A and the
medial axis of A can be defined as in the case of (usual) planar shapes.
In particular, ∂A derives from ∂A by disconnecting the latter boundary at
the contact points u and v of the splitting disk D, and reconnecting it with
the circular arcs uv and vu. See Figure 5.10, which shows the boundary of
a domain augmented with two disks, one for each of its two holes (i.e., the
sites s1 and s2 ).
Concerning the medial axis, every maximal inscribed disk in A different
from D corresponds to exactly one maximal inscribed disk in A . The medial
axis of A therefore is the same geometric graph as M (A), except that the
edge of M (A) containing the center of D is split into two disconnected
edges which both have the center of D as one of their endpoints. These two
points are two leaves of M (A ).
To draw the connection to the edge graph GS of V (S), the initial
shape B from above is augmented with |S| − 1 maximal inscribed disks,
s1
s2
Figure 5.10.
Oriented boundary of an augmented domain (from [25]).
70
Chapter 5. Generalized Sites
namely, the ones centered at the points q(s) ∈ GS , where q(s) was the
vertical projection onto GS of a point with smallest y-coordinate on the
site s. Denote by AS the resulting domain after these |S| − 1 augmentation
steps. We may conclude the main finding of this subsection as follows.
Lemma 5.3. The tree TS = GS \{q(s) | s ∈ S\{Γ}} is the medial axis of
the augmented domain AS .
From the algorithmic point of view, augmenting B amounts to
connecting its boundary ∂B to a single cyclic sequence, ∂AS , that
consists of pieces from ∂B and from circles bounding the splitting disks.
(One-dimensional sites contribute to ∂B with two curves, one for either
orientation, and the special case of point sites can be handled consistently.)
Each such boundary piece is used exactly once on ∂AS , and traversing ∂AS
corresponds to tracing the medial axis tree M (AS ) in preorder (i.e., root
first, then the subtrees for its children recursively, in radial order).
Of course, the construction of ∂AS is trivial once the splitting disks
are available. The non-trivial task is to find these disks Di , one for each
site si ∈ S\{Γ}. Recall that Di is horizontally tangent to si at a lowest
point p(si ) of si . The center q(si ) of Di lies on the edge graph GS of V (S)
but, of course, Di needs to be found without knowledge of GS . Indeed, a
simple and efficient plane sweep can be applied in O(n log n) time if the
sites in S are described by a total of n objects, each being manageable in
constant time.
Note that ∂AS then consists of Θ(n) pieces, that either bound splitting
disks, or pieces that stem from site boundaries. The former pieces are used
only to link the site segments in the correct cyclic order. They do not play
any geometric role, and can be ignored when the medial axis algorithm of
Subsection 5.5.2 is applied to AS in order to compute the desired Voronoi
diagram V (S).
5.5.2. Medial axis algorithm
We now describe the medial axis algorithm for circular arc shapes in [27, 34]
that is suited to our needs. This algorithm works without modifications for
the augmented domains introduced in Subsection 5.5.1.
Let A be some planar shape bounded by n circular arcs. We restrict
attention to circular curves, because they are well suited to approximate
more general objects accurately (for example, objects bounded by spline
curves), and the case of line segments is trivially covered. Moreover,
if the approximation by circular arcs (in particular, biarcs, i.e., smooth
concatenations of two circular arcs) is done carefully, stability of the medial
5.5. Voronoi diagrams for curved objects
71
axis can be guaranteed [34]. For a recent survey on medial axes for general
objects, the interested reader is referred to Attali et al. [83].
For the shape A, call a disk D ⊆ A maximal if there exists no disk D
different from D such that D ⊃ D and D ⊆ A holds. Then the medial axis,
M (A), of A can be defined as the (infinite) set of all centers of maximal
disks for A. Observe that this definition is equivalent to the distance-based
definition given in Subsection 5.5.1. The corresponding infinite set of disks
(its union gives A) is commonly called the medial axis transform of A.
Sometimes also the term symmetric axis is used to denote the medial axis
(transform) of a shape.
M (A) is connected and cycle-free and thus forms a tree. It consists of
O(n) edges, which are maximal pieces of straight lines and (possibly all
four types of) conics. Endpoints of edges will be called vertices of M (A).
Compared to polygonal shapes, the medial axis for circular arc shapes is not
more complicated, as both structures contain edges of algebraic degree 2 in
general; see Figure 5.11.
We give a simple and practical divide & conquer randomized algorithm
for computing M (A). The costly part is delegated to the divide step, which
basically will consist of inclusion tests for arcs in circles. The merge step is
trivial; it just concatenates two partial medial axes. The expected runtime
is bounded by O(n3/2 ), and can be proven to be O(n polylog n) for several
types of shape.
Figure 5.11.
Medial axis of a shape bounded by circular arcs.
Chapter 5. Generalized Sites
72
A qualitative difference to existing medial axis algorithms is that a
combinatorial description of M (A) is extracted first, which can then be
directly (and robustly) converted into a geometric representation. The
algorithm is based on the following simple though elegant decomposition
lemma in Choi et al. [219].
Lemma 5.4. Consider any maximal disk D for A. Let A1 , . . . , At be the
connected components of A\D, and denote by p the center of D.
(1)
M (A) =
t
M (Ai ∪ D),
i=1
(2)
{p} =
t
M (Ai ∪ D).
i=1
That is, using any maximal disk one can compute the medial axes for the
resulting components recursively, and then glue them together at the disk’s
center. By a moving argument, a balanced decomposition always exists.
Lemma 5.5. There exists a maximal disk D for A such that at most
arcs from ∂A are (completely) contained in each component of A\D.
n
2
For an edge e of M (A), define Walk(e) as the path length in M (A)
from e to p∗ , the center of a balanced disk as in Lemma 5.5. Further,
define Cut(e) as the size of the smaller one between the two subtrees which
constitute M (A)\{e}. Any tree with small ‘cuts’ tends to have short ‘walks’,
in the following respect.
Lemma 5.6. If an edge e of M (A) is chosen uniformly at random, then
we have E[Walk(e)] = Θ(E[Cut(e)]).
Lemma 5.6 motivates the disk finding algorithm below, which combines
random cutting with local walking. Its main subroutine, MAX(b), selects
for an arc b ⊂ ∂A its midpoint x and returns the unique maximal disk for A
with x on its boundary. Let c ≥ 3 be a (small) integer constant.
Procedure CUT(A)
Put A = A
Repeat
Choose a random arc b of ∂A
Compute D=MAX(b) and let A0 be the larger
component of A induced by D
Assign A = A ∩ A0
Until A0 contains less than n − nc arcs
Report D
5.5. Voronoi diagrams for curved objects
73
Procedure WALK(A)
Choose a random arc b of ∂A
Compute D=MAX(b)
Let A0 be the larger component induced by D
While A0 contains more than n − nc arcs do
Let b1 (b2 ) be the first (last) complete arc of ∂A in A0
Compute D1 =MAX(b1 ) and D2 =MAX(b2 )
Assign to A0 the smaller one of the respective larger
components of A for D1 and D2
Memorize the corresponding disk D ∈ {D1 , D2 }
Report D
The disk finding algorithm now combines the CUT procedure and the
WALK procedure as follows. The repeat loop of CUT and the while loop of
WALK are executed by turns. Whenever CUT is closer to the goal (i.e.,
yields a smaller largest component than does WALK), we readjust the
current disk for WALK to be that of CUT. Termination takes place in
√
either WALK or CUT. Using Lemma 5.6, a bound of O( n) on the expected
number of total loop executions in CUT and WALK can be proven.
The costly part in both procedures is their subroutine MAX, whose
√
expected number of calls obeys the same bound, O( n). Computing
D = MAX(b) has a trivial implementation which runs in O(n) time: We
initialize the disk D as the (appropriately oriented) halfplane that supports
b at its midpoint x. Then, for all remaining arcs bi ⊂ ∂A that intersect D,
we shrink D so as to touch bi while still being tangent to b at x. The most
complex operation for shrinking D is computing the intersection of two
circles. In particular, and unlike previous medial axis algorithms, no conics
take part in geometric operations.
The randomized complexity for computing the medial axis is thus given
by the recurrence relation
1
1
·n +T
1−
· n + O(n3/2 ) = O(n3/2 ).
T (n) = T
c
c
In many cases, however, the algorithm will perform substantially better.
Let ∆ be the graph diameter of M (A) (i.e., the maximal number of edges
on a path in this tree). If ∆ = Θ(log n), which is the smallest value possible,
then an overall runtime of O(n log2 n) is met. For the other extreme case,
∆ = Θ(n), our strategy is even faster, O(n log n). The latter situation is
quite relevant in practice, because an input shape, even if not branching
much, is typically approximated by a large number of circular arcs.
Notice that the algorithm works exclusively on the boundary ∂A of A,
except for a final step, where the conic edges of M (A) are explicitly
Chapter 5. Generalized Sites
74
(a) Point sites
(b) Mixed sites
Figure 5.12. Voronoi diagrams for different kinds of sites, computed by the generalized
medial axis algorithm (from [25]).
calculated and reassembled. This gives rise to increased numeric stability
in comparison to other existing approaches. The necessary implementation
details, including a treatment of degenerate situations, and a discussion of
the base cases occurring in this divide & conquer algorithm, are given in
the paper [27]. Figures 5.12(a) and 5.12(b) give two examples of the output.
We remark at this place that approximating general objects by circular
arcs — instead of line segments — drastically reduces the input volume,
2
namely from N line segments to O(N 3 ) = n circular arcs, for fixed
accuracy ε; see e.g. [34]. This is particularly desirable in applications
concerning motion planning and offset calculation (a standard operation
in image processing).
The δ-offset of a shape A is the Minkowski sum of A and a disk with
radius δ. Whereas the class of polygons is not stable under taking offsets
(because the δ-offset of a polygon will also contain circular arcs in its
boundary), the class of circular arc shapes does have this property. This
is useful, for example, in computer-aided manufactoring; see Held et al.
[400] and Seong et al. [635]. In fact, the medial axis algorithm presented
in this subsection delivers a link structure for ∂A that allows for offset
computations without constructing M (A) explicitly; see [25].
A different possibility for (polygon) offsetting, the so-called mitered
offset, results from the straight skeleton of a polygon, discussed in
Section 5.3.
Chapter 6
HIGHER DIMENSIONS
An important generalization of the Voronoi diagram for point sites is to
Euclidean d-space Rd , for d ≥ 3. Naturally the three-dimensional space —
the space we live in — holds a predominant role, especially concerning
applications in the natural sciences. But also higher dimensions are strongly
relevant, either in direct applications like clustering multivariate data, or
indirectly, like embedding a generalized Voronoi diagram into a polyhedral
cell complex of larger dimension, for analytic or algorithmic purposes.
Several nice properties are retained in higher dimensions (e.g., the
convexity of the regions, and with it, the piecewise linear structure), while
others are lost (e.g., the linear worst-case size of the diagram).
Voronoi diagrams in d-space are closely related to geometric objects
in different dimensions. These relationships, and their structural and
algorithmic implications, are also discussed in this chapter.
6.1. Voronoi and Delaunay tessellations in 3-space
6.1.1. Structure and size
Let us consider the classical Voronoi diagram in R3 first. Let S be a set of n
point sites in 3-space. The bisector of two sites p, q ∈ S is the perpendicular
plane through the midpoint of the line segment pq. The region VR(p, S) of a
site p ∈ S is the intersection of halfspaces bounded by bisectors, and thus is
a three-dimensional convex polyhedron. The boundary of VR(p, S) consists
of facets (maximal subsets within the same bisector), of edges (maximal line
segments in the boundary of facets), and of vertices (endpoints of edges).
The regions, facets, edges, and vertices of V (S) define a polyhedral cell
complex in 3-space. This cell complex is face-to-face: If two regions have a
non-empty intersection f , then f is a face (facet, edge, or vertex) of both
regions.
75
76
Chapter 6. Higher Dimensions
As an appropriate data structure for storing a three-dimensional
cell complex we mention the facet-edge data structure in Dobkin and
Laszlo [284].
The number of facets of VR(p, S) is at most n − 1; at most one for each
site q ∈ S \ {p}. Hence, by the Eulerian polyhedron formula, the number
of edges and vertices of VR(p, S) is O(n), too. This shows that the total
number of components of the diagram V (S) in 3-space is O(n2 ). In fact,
there are configurations S that force each pair of regions of V (S)
to share
n
a facet, thus achieving their maximum possible number of 2 ; see, e.g.,
Dewdney and Vranch [268].
This fact sometimes makes Voronoi diagrams in R3 less useful
compared to those in the plane. On the other hand, Dwyer [296] showed
that the expected size of V (S) in Rd is only O(n) for constant dimensions d,
provided S is drawn uniformly at random in the unit ball. The analogous
result was shown by Bienkowski et al. [131] for points drawn from a ddimensional cube. These results indicate that high-dimensional Voronoi
diagrams will be small in many practical situations (Figure 6.1).
In analogy to the two-dimensional case, the Delaunay tessellation
DT(S) in 3-space is defined as the geometric dual of V (S). It contains
a tetrahedron for each vertex, a triangle for each edge, and an edge for
each facet, of V (S). Equivalently, DT(S) may be defined using the empty
sphere property, by declaring a tetrahedron spanned by S as Delaunay iff
its circumsphere is empty of sites in S. The centers of these empty spheres
are just the vertices of V (S).
DT(S) is a partition of the convex hull of S into tetrahedra, provided S
is in general position (no four points coplanar, no five points co-spherical),
which will be assumed in the sequel. Note that, for special configurations of
point sites in 3-space, the edges of DT(S)
may already form the complete
graph on S, i.e., the graph where all n2 vertex pairs are matched with an
edge. The cell complex DT(S) is also called the Delaunay tetrahedrization
or the Delaunay simplicial complex for S.
From the many optimization properties of planar Delaunay triangulations (described in detail in Section 4.2), only a few find their generalizations in three and higher dimensions. As shown in Rajan [598], one of
them is coarseness, that is, DT(S) minimizes the largest circumsphere
of its tetrahedra, among all possible tetrahedrizations of the point set S.
(Simultaneously, also the largest minimum enclosing sphere is minimized.)
Both concepts of coarseness are equivalent for so-called self-centered
tetrahedrizations (or more generally, simplicial complexes in d-space),
where each tetrahedron (d-simplex) is required to contain its circumcenter.
6.1. Voronoi and Delaunay tessellations in 3-space
Figure 6.1.
77
A cube is split into various Voronoi regions.
As an interesting fact, each self-centered simplicial complex is necessarily a
Delaunay tessellation [598, 618].
Regrettably, the important and useful equiangularity property is lost in
dimensions d ≥ 3; see Schmitt and Spehner [618]. DT(S) does not maximize
the smallest occurring angle in its tetrahedra, for none of the direct threedimensional generalizations of an angle. Still, a certain dual optimization
property for angles is preserved [618].
6.1.2. Insertion algorithm
Various methods for constructing the Voronoi diagram V (S) in 3-space
have been proposed; they are surveyed to some extent in [93]. Among them,
78
Chapter 6. Higher Dimensions
incremental insertion of sites (cf. Section 3.2) is most intuitive and easy to
implement.
Basically, two different techniques for integrating a new site p into V (S)
have been applied. The more obvious method first determines all facets
of the region of p in the new diagram, V (S ∪ {p}), and then deletes the
parts of V (S) interior to this region; see e.g. Watson [699], Field [340], and
Tanemura et al. [678]. Inagaki et al. [422] describe a robust implementation
of this method.
In the dual environment, this amounts to detecting and removing all
tetrahedra of DT(S) whose circumspheres contain p, and then filling the
‘hole’ with empty circumsphere tetrahedra with p as an apex, to obtain
DT(S ∪ {p}).
Joe [430], Rajan [598], and Edelsbrunner and Shah [313] follow a
different and numerically more stable approach. Like in the planar case,
after having added a site to the current Delaunay tessellation, certain flips
(described below) that change the local tetrahedral structure are performed
in order to restore local Delaunayhood. The existence of such a sequence
of flips is less trivial, however; see Section 6.3.
Let us focus on a single flip. Recall that in R2 there are exactly two
ways of triangulating four sites in convex position, and that a flip changes
one into the other. In R3 there are also two ways of tetrahedrizing five
sites in convex position, and a flip per definition exchanges them. (Such a
flip is also called a bistellar flip.) Note, however, that the flip will replace
two tetrahedra by three or vice versa; see Figure 6.2. This indicates an
important difference between triangulations and tetrahedral tessellations:
The number of tetrahedra does depend on the way of tetrahedrizing S. It
may vary from Θ(n) to Θ(n2 ); see Section 6.3 for more details.
Figure 6.2.
The 2-to-3 tetrahedra flip in 3-space.
6.1. Voronoi and Delaunay tessellations in 3-space
79
Each triangle ∆ of the tessellation (except for convex hull triangles)
is shared by two tetrahedra T and T which, in turn, are spanned by five
sites. Three of them span ∆, and the remaining two sites q and q belong
to T and T , respectively. The triangle ∆ is called locally Delaunay if the
circumsphere of T does not enclose q (or, equivalently, if the circumsphere
of T does not enclose q). A third tetrahedron might be spanned by these
five sites. ∆ is called flippable if the union of these two or three tetrahedra
is convex.
After having added a site p ∈ S to the current Delaunay tessellation,
the algorithm first splits the tetrahedron containing p into four tetrahedra
with apex p, in the obvious way. The four triangles opposite to p are then
considered, which are the bases of the tetrahedra with apex p. For each such
triangle ∆, a flip is performed that involves the five sites corresponding to ∆,
provided ∆ is flippable and not locally Delaunay. Thereby, new triangles
become opposite to p and possibly have to be flipped, too. This sequence
of flips terminates with the Delaunay tessellation that includes p.
The algorithm works locally and thus is elegant and relatively easy to
understand. It runs in worst-case optimal O(n2 ) time. Still, the incremental
approach has its drawbacks. It might construct intermediate tessellations of
quadratic size, in spite of a possible linear size of the final tessellation. This
unpleasant phenomenon even cannot be overcome by inserting the sites
in random order. Proving the existence of an — in this sense — efficient
insertion order is an open problem. See Snoeyink and van Kreveld [660] for
a related result in the plane.
The algorithm can be found in more detail in Edelsbrunner and
Shah [313], including correctness proofs and a data structure for storing
tetrahedral tessellations. Joe [431] provides an efficient implementation in
3-space, and Cignoni et al. [223] propose a hybrid method that works
in general d-space and efficiently combines insertion, divide & conquer,
and bucketing. Rajan [598] and Edelsbrunner and Shah [313] discuss the
d-dimensional variant of the incremental insertion algorithm, and the
latter paper also provides a generalization to so-called regular simplicial
tessellations. Flips and other material on such regular simplicial cell
complexes will be discussed in some detail in Section 6.3.
6.1.3. Starting tetrahedron
An important issue we have ignored so far in our discussion: Finding a
starting tetrahedron that contains a newly inserted site. Such a tetrahedron
has to be identified before the process of restructuring the tessellations
can take place. Interestingly, an efficient search may be based on the
80
Chapter 6. Higher Dimensions
construction history of the tessellation. See Edelsbrunner and Shah [313],
who describe a d-dimensional generalization of the strategy for planar
Delaunay triangulations in Guibas et al. [390]. This strategy is similar
in spirit to the method of conflicting triangles described in Sections 3.2
and 6.5.
From the practical point of view, conceptually simpler techniques
may be preferable, even though they are theoretically inferior. Bucketing
techniques like the one implemented in Joe [431] may be used. As another
alternative, we may simply ‘walk’ in the tessellation from some chosen point
q to the target point p (the site to be inserted). Walking strategies work for
arbitrary tetrahedral tessellations and have been systematically studied in
Devillers et al. [266]. We briefly review their findings below.
The straight walk passes through all the tetrahedra along the line
segment qp, whereas the orthogonal walk visits the tetrahedra along an
isothetic (i.e., axis-parallel) path, moving from q to p by changing only
one coordinate at a time. Visibility walk, on the other hand, moves from
the current tetrahedron t to some tetrahedron adjacent to t, through
a facet (triangle) ∆ if the plane supporting ∆ separates t from p.
More than one facet may fulfill this condition, and choice may be done
randomly (stochastic visibility walk). For Delaunay tessellations and regular
tessellations (Section 6.3), visibility walk always reaches the correct target.
This is a consequence of the face acyclicity property with respect to the infront/behind relation of these structures, described in De Floriani et al. [245]
and in Edelsbrunner [301]; see Section 4.2. In the case of an arbitrary
tessellations, visibility walk may in fact loop, though stochastic visibility
walk will terminate with high probability.
The expected number of triangles visited by straight walk in a planar
√
Delaunay triangulation of uniformly distributed points is O(d(q, p) n),
where d(q, p) denotes the distance between q and p; see Bose and
Devroye [158] and Devroye et al. [267]. In the worst case in 3-space, however,
this walk may have to pass through Θ(n2 ) tetrahedra. The best choice is the
stochastic visibility walk, in spite of its extreme behavior in the worst case.
It performs experimentally slightly better than the straight walk and the
orthogonal walk, is easier to code, and does not encounter problems with
geometric degeneracies where, for example, the line segment qp intersects
an edge or vertex of a tetrahedron.
Of course, choosing a ‘good’ starting point q is an issue, no matter
which kind of walk is used. An efficient technique for selecting q in two- and
three-dimensional Delaunay tessellations is described by Mücke et al. [551].
Their point-location method for walking to the target point p also does not
require any preprocessing of the tessellation.
6.2. Power diagrams
81
6.2. Power diagrams
Voronoi diagrams are intimately related to geometric objects in higher
dimensions. This fact, along with one of its algorithmic applications, has
already been addressed in Section 3.5. Here, we base the discussion on a
generalization of Voronoi diagrams called power diagrams; the geometric
correspondences to be described extend to that type in a natural manner.
We refer to d dimensions in order to point out the general validity of the
results.
6.2.1. Basic properties
Consider a set S of n point sites in d-space Rd . Assume that each point
in S has assigned an individual weight w(p). In some sense, w(p) measures
the capability of p to influence its neighborhood. This is expressed by the
so-called power function of a point x ∈ Rd with respect to a site p ∈ S,
pow(x, p) = (x − p)T (x − p) − w(p).
A nice geometric interpretation is the following. For positive weights,
a
weighted site p can be viewed as a sphere with center
p and radius w(p);
for all points x outside this sphere, pow(x, p) > 0, and pow(x, p) expresses
the length of a line segment starting from x and tangent to the sphere.
The locus of equal power with respect to two weighted sites p and q is
a hyperplane, called the power hyperplane of p and q. (In the plane, the
power line of two circles is also called their chordale or radical axis. It passes
through their points of intersection, if any, because the power function
vanishes there for both circles.) Let h(p, q) denote the closed halfspace
bounded by this hyperplane and containing the points of less power with
respect to p. Then the power cell of p is given by
h(p, q).
cell(p) =
q ∈ S\{p}
In analogy to the classical Voronoi regions, the power cells define a partition
of Rd into convex polyhedra, the so-called power diagram, PD(S), of S. Also
the names Laguerre-Voronoi diagram or generalized Dirichlet tessellation
are used in the literature; see e.g. Imai et al. [420]. Figure 6.3 depicts a
planar example.
PD(S) coincides with the Voronoi diagram of S if all weights are the
same, or even are zero. In contrast to Voronoi regions, however, power cells
might be empty if general weights are used; for instance, cell(p) = ∅ in
Figure 6.3. A condition on weights sufficient for making each site remain
within its power cell is stated in Section 10.5. Note that adding a fixed
Chapter 6. Higher Dimensions
82
p
Figure 6.3.
Power diagram for 10 circles in the plane.
constant ∆ to all weights leaves the power diagram unchanged. So insisting
on w(p) ≥ 0 for all p ∈ S would cause no loss of generality.
PD(S) is a face-to-face cell complex in d-space that consists of
polyhedral faces of various dimensions
d+1j, for 0 ≤ j ≤ d. In the nondegenerate case, exactly d + 1 edges, 2 facets (faces of dimension d − 1),
and d + 1 cells (the power cells) meet at each vertex of PD(S). Such
complexes are commonly called simple cell complexes in Rd . For storing
a d-dimensional cell complex, the cell-tuple data structure in Brisson [168]
seems appropriate. This data structure represents the incidence and
ordering information in a cell complex in a simple uniform way.
Wheneach (positively) weighted site p ∈ S is interpreted as the sphere
σp = (p, w(p)), we can make the following nice observation. The part
of σp that contributes to the union of all these spheres, p ∈ S σp , is just
the part of σp within cell(p). This means that PD(S) defines a partition of
this union into simply-shaped and algorithmically tractable pieces. Several
algorithms concerning the union (and also the intersection) of balls are
based on this partition; see Avis et al. [107], Aurenhammer [90], and
Edelsbrunner [302]. For the intersection of balls, their order-(n − 1) power
diagram (Subsection 6.5.2) is used.
6.2. Power diagrams
83
6.2.2. Polyhedra and convex hulls
Power diagrams (and thus Voronoi diagrams) are, in a strong sense,
equivalent to the boundaries of convex polyhedra in one dimension higher.
This is a fact with far-ranging implications and has been observed in
Brown [169], Klee [460], Edelsbrunner and Seidel [310], Paschinger [586],
and Aurenhammer [87]. The power function pow(x, p) in Rd can be
expressed by the hyperplane
π(p) : xd+1 = 2xT p − pT p + w(p)
in Rd+1 , in the sense that a point x lies in cell(p) of PD(S) iff,
at x, π(p) is vertically above all hyperplanes π(q) for q ∈ S \ {p}.
Hence PD(S) corresponds, by vertical projection, to the upper envelope
of these hyperplanes, which is the surface of an unbounded convex
polyhedron, Π(S), in Rd+1 . Conversely, it is not difficult to see that every
upper envelope of n non-vertical hyperplanes in Rd+1 corresponds to the
power diagram of n suitably weighted sites in Rd .
Cell complexes that can be obtained from projecting the boundary
of a convex polyhedron (onto one of its facets) are also called Schlegel
diagrams in convex geometry. We conclude that power diagrams and
Schlegel diagrams are the same structures, up to clipping the unbounded
faces of the power diagram. For a wealth of material on convex polytopes
and cell complexes, and on convex geometry in general, the interested reader
is referred to the monograph by Grünbaum [384] and to the handbook by
Gruber and Wills [383].
The following upper bound is a direct consequence of the upper bound
theorem for convex polyhedra proved in McMullen [529]. The bound
is trivially sharp for power diagrams, but is achieved also for Voronoi
diagrams, as was shown in Seidel [625]. A respective construction of a point
configuration S in d-space is sketched at the beginning of Chapter 10.
Theorem 6.1. Let S be a set of n point sites in Rd . Any power diagram
for S, and in particular, the Voronoi diagram for S, realizes at most fj faces
of dimension j, for
fj =
a i
n−d+i−2
i=0
j
where a = d2 and b =
i
d
2
+
b d−i+1 n−d+i−2
i=0
j
i
,
. The numbers fj are O(n 2 ), for 0 ≤ j ≤ d−1.
d
84
Chapter 6. Higher Dimensions
For algorithmic issues, power diagrams in Rd can be brought in
connection to convex hulls in Rd+1 , by exploiting a duality transform
(actually, polarity) between upper envelopes of hyperplanes (or intersections
of upper halfspaces) and convex hulls of points. This connection is best
described by generalizing the lifting map in Section 3.5 to weighted points.
A site p ∈ S with weight w(p) is transformed into the point
λ(p) =
p
pT p − w(p)
in Rd+1 . There is a interrelation called polarity between the transforms λ
and π. The point λ(p) is called the pole of the hyperplane π(p) which,
in turn, is called the polar hyperplane of λ(p). Polarity defines a oneto-one correspondence between arbitrary points in Rd+1 and non-vertical
hyperplanes in Rd+1 . It is well known that polarity preserves the relative
position of points and hyperplanes.
In particular, if w(p) = 0, then point λ(p) lies on the surface of a
paraboloid P in Rd+1 whose axis of revolution is the xd+1 -axis, and the
hyperplane π(p) is tangent to P at λ(p); cf. Section 3.5 for the threedimensional case.
To show the connection to convex hulls, consider an arbitrary face f of
the unbounded polyhedron Π(S) defined above. Let f be the intersection of
m = d − j + 1 hyperplanes π(p1 ), . . . , π(pm ), such that f is of dimension j.
Each point x ∈ f lies on these but above all other hyperplanes π(q) defined
by S. Hence, the polar hyperplane of x has the points λ(p1 ), . . . , λ(pm )
on it and the remaining points λ(q) above it. This shows that the points
λ(p1 ), . . . , λ(pm ) span a face of dimension d − j of the lower convex hull of
the point set {λ(p) | p ∈ S}.
We conclude that each j-dimensional face of Π(S), and thus of PD(S), is
represented by a (d − j)-dimensional face of this convex hull. This implies
a duality between power diagrams in Rd and convex hulls (i.e., convex
polytopes) in Rd+1 .
In the special case of an unweighted point set S in the plane, the parts
of the convex hull that are visible from the plane project to the vertices,
edges, and triangles of the Delaunay triangulation of S, and we obtain
Theorem 3.5 of Section 3.5. A triangulation which can be obtained by
(vertically) projecting the boundary faces of some convex hull is called a
regular triangulation; see, e.g. Lee [491] and Edelsbrunner and Shah [313].
Regular triangulations are just those being dual to planar power diagrams.
We will provide some material on regular simplicial complexes in general
dimensions in Section 6.3.
6.2. Power diagrams
85
Once the convex hull of the point set {λ(p) | p ∈ S} has been computed,
the faces of PD(S), as well as their incidence and ordering relations, can be
obtained in time proportional to the size of PD(S).
Theorem 6.2. Let Cd+1 (n) be the time needed to compute a convex hull of
n points in Rd+1 . A power diagram (and in particular, the Voronoi diagram)
of a given n-point set in Rd can be computed in Cd+1 (n) time.
Worst-case optimal convex hull algorithms working in general
dimensions have been designed by Clarkson and Shor [229], Seidel [630],
d
and Chazelle [194], yielding Cd+1 (n) = O(n log n + n 2 ). So Theorem 6.2
is asymptotically optimal in the worst case. Note, however, that power
diagrams in higher dimensions may as well have a fairly small size, O(n),
which emphasizes the use of output-sensitive convex hull algorithms. The
algorithm in Seidel [627] achieves Cd+1 (n) = O(n2 + f log f ), where f
is the total number of faces of the convex hull constructed. The latest
achievements are C4 (n) = O((n + f ) log2 f ) in Chan et al. [191] and
C5 (n) = O((n + f ) log3 f ) in Amato and Ramos [55]. Mehlhorn et al. [535]
provide a kernel for higher-dimensional geometric computation and describe
its application to convex hulls and Delaunay triangulations.
A thorough and detailed overview of the existing convex hull algorithms
for various dimensions is provided in Seidel [633].
Among the possible direct methods that construct the power diagram
in R2 without resorting to one more dimension are randomized incremental
insertion (see Section 3.2) and divide & conquer (see Imai et al. [420]).
Both run in optimal time, O(n log n). The linear-time insertion method
for power cells in Aurenhammer and Edelsbrunner [96] yields a worst-case
optimal O(n2 ) algorithm in R3 .
6.2.3. Related diagrams
Space constraints preclude our discussion of power diagrams. Still, some
remarks are in order to point out their central role within the context of
Voronoi diagrams. For a more detailed discussion of the following material,
and references, see Aurenhammer and Imai [98].
The regions of a Voronoi diagram are defined by a set of sites and a
distance function. If the regions are polyhedral, then any Voronoi diagram
defined in this way can be shown to be the power diagram of some suitable
set of weighted point sites. For instance, this is the case for the farthest
site Voronoi diagram, whose regions consist of all points having the same
farthest site in the given set. In fact, any Voronoi diagram of higher order
(see Section 6.5) is a power diagram.
86
Chapter 6. Higher Dimensions
Note that the feature of regions being polyhedral is equivalent to
the feature that all bisectors are hyperplanes, and thus the latter is a
characterizing property of power diagrams as well. This fact turns out
useful, for example, in the study of Voronoi diagrams in parametric
statistical spaces, where distances between points measure divergences
between distributions. We refer to Sadakane et al. [614] for the Kullback–
Leibler divergence, and to Boissonnat et al. [145] for a study of general
Bregman divergences. In particular, certain hyperbolic Voronoi diagrams
(see Subsection 7.1.1) that arise in this context can be interpreted as power
diagrams.
Voronoi diagrams all of whose separators are hyperplanes are also called
affine Voronoi diagrams. Such diagrams can always be generated by general
quadratic-form distances; see Subsection 7.4.5 for a short discussion of some
interesting instances.
A polyhedral cell complex in Rd is called simple if exactly d + 1
cells meet at each vertex. For example, the Voronoi diagram of a set of
point sites in Rd is simple if no d + 2 sites are co-spherical. If d ≥ 3,
any simple cell complex can be shown to be a power diagram. Note that,
interestingly, for d = 2 this property is lost. An intuitive reason is that
polygonal partitions of the plane allow plenty of freedom, whereas forming
a valid cell complex in higher dimensions comes with several geometric
restrictions.
The class of power diagrams is closed under taking cross-sections with
a hyperplane. That is, the diagram obtained from intersecting a power
diagram in Rd with some hyperplane is again a power diagram, in Rd−1 . As
a seemingly unrelated application of this fact, certain linear combinations
between a weighted site and its neighbors in the power diagram can
be obtained, which are of use in smoothing surfaces based on Delaunay
triangulations and regular triangulations; see Section 4.2.
Several generalized Voronoi diagrams in Rd have an embedding in
a power diagram in Rd+1 , in the sense that they can be obtained by
intersecting a power diagram with simple geometric objects (like cones,
spheres, or paraboloids), and then projecting this intersection.
For example, the additively weighted Voronoi diagram (i.e., the
Voronoi diagram for spheres, or the Johnson–Mehl model [433]), and the
multiplicatively weighted Voronoi diagram [96, 615] (or the Apollonius
model ) have this property. Weighted Voronoi diagrams tend to have
complex geometric and combinatorial properties, especially in higher
dimensions, and their embeddings in power diagrams can be viewed as
linearizations at the expense of adding one more dimension. We will come
back to weighted models of Voronoi diagrams and their applications in
6.3. Regular simplicial complexes
87
Section 7.4. The Voronoi diagram for spheres is also briefly treated at the
end of Section 6.6.
In all situations mentioned above, a set of weighted sites for the
corresponding power diagram can be computed easily. Thus general
methods of handling Voronoi diagrams and cell complexes in d-space
become available. For example, the Voronoi diagram for spheres in 3-space,
and the multiplicatively weighted Voronoi diagram in the plane, can both be
computed in O(n2 ) time, which is asymptotically optimal in the worst-case.
6.3. Regular simplicial complexes
A polyhedral cell complex in d-space Rd is called simplicial if all its cells
are simplices. A k-simplex in Rd (for 0 ≤ k ≤ d) is the convex hull of
k + 1 affinely independent points. For example, 3-simplices are tetrahedra,
2-simplices are triangles, 1-simplices are line segments, and 0-simplices are
just points. The boundary of a simplex consists of various simplices of
lower dimensions. Cell complexes that are dual to simplicial complexes are
necessarily simple cell complexes, and vice versa.
For instance, the classical Voronoi diagram (respectively, the power
diagram) in Rd are simple cell complexes, if sites (respectively, weights)
are chosen sufficiently general, implying that the Delaunay complex is then
simplicial.
While the numbers of edges and triangles in any triangulation are fixed
with a given configuration S of its n vertices in the plane (see Section 2.2),
the size of simplicial complexes in higher-dimensional space Rd may vary
d
largely, from Θ(n) to Θ(n 2 ). See Rothschild and Straus [610] for the
upper bound, which is already attained by Delaunay simplicial complexes;
cf. Section 10.1. Note that this is also the asymptotic maximum size of a
convex hull in Rd+1 , by the upper bound theorem in Subsection 6.2.2.
Edelsbrunner et al. [312] showed that, for any n-point set in R3 , there
is a tetrahedrization consisting of only 3n − 11 tetrahedra or less. Very
recently, this was generalized to d ≥ 4 by Aichholzer et al. [40], who showed
a number of d-simplices of at most dn + Ω(log n). On the other hand, if we
ask for the largest-sized tetrahedrization every n-point set in R3 (in general
5
position) has to admit, the best known bound is O(n 3 ); see Brass [163].
The fact that every point set in Rd has a small simplicial complex
suggests the idea of ‘slimming’ the Delaunay simplicial complex DT(S),
by adding point sites that lie in many Delaunay spheres. As a surprising
result, Bern et al. [128] showed that only O(n) extraneous points suffice to
achieve a size of O(n) in general dimensions d, improving the bound given
in Chazelle et al. [198].
88
Chapter 6. Higher Dimensions
6.3.1. Characterization
So-called regular simplicial complexes are a subclass well studied in convex
geometry. As mentioned in Section 6.1, these complexes are intimately
related to convex hulls and power diagrams, by projection and duality,
respectively. Algorithmic applications arise, for example, from certain
subcomplexes introduced as weighted α-shapes in Edelsbrunner [302], a
concept useful for describing finite unions of balls, and thus, for the
approximate shape and medial axis description of solids; see Section 6.6.
Also, certain optimization properties of Delaunay triangulations described
in Sections 4.2 and 6.1 generalize to regular simplicial complexes. We review
some more of their properties in the sequel.
Let S be a set of n point sites in Rd , and assume each site p ∈ S
has assigned some real weight w(p). Like the Delaunay tessellation can
be defined by the empty-circle or empty-sphere property (see Sections 4.2
and 6.1), we can define a regular simplicial complex for weighted sites by
generalizing this property.
Consider two weighted sites p and q in Rd , not necessarily in S, with
weights w(p) and w(q). We call p and q orthogonal (obtuse, acute) if d(p, q)2
is equal, larger, or smaller than w(p) + w(q), respectively.
Notationstems
from interpreting these two sites as spheres, with radii w(p) and w(p)
if w(p), w(q) ≥ 0 (see Section 6.2), which intersect at a respective angle.
Now, consider d + 1 weighted sites in the set S above. Their convex
hull is a d-simplex ∆. It is easy to show that there exists a unique sphere σ
orthogonal to each of these d + 1 weighted sites. (If their weights are zero
then σ is just the circumsphere of ∆.) Simplex ∆ is called regular (in S) if
all other weighted sites in S are obtuse to σ. See Figure 6.4 for an example
in the plane.
The collection of all regular d-simplices for the weighted point set S
defines a simplicial complex which resides in the convex hull conv(S) of S,
and which is called a regular tessellation of S.
Not every simplicial complex is regular. Already for dimension d = 2,
well-known examples of triangulations exist which are not regular; see
e.g. [88, 300].
By the geometric relations presented in Section 6.2, we have the
following characterization.
Theorem 6.3. Let S be a set of n point sites in Rd , and let Σ be a
simplicial complex in conv(S) whose vertex set is a subset of S. The
following statements are equivalent:
(a) Σ is the regular tessellation of S, for some weights for S.
(b) Σ is dual to the power diagram of S, for some weights for S.
6.3. Regular simplicial complexes
89
σ
Figure 6.4. A regular 2-simplex (triangle). The dashed circle σ orthogonally intersects
the three weighted sites (circles) centered at the triangle vertices, and is obtuse to the
remaining two weighted sites.
(c) Σ is the vertical projection of the lower boundary part of the convex hull
of a point set S in R(d+1) , obtained from S by assigning some (d+1)th
coordinates.
6.3.2. Polytope representation in weight space
Given a set S of sites, a variety of regular tessellations exists, depending on
the weight vector W we assign to S. Certain sites in S may be missing as
vertices in the tessellation, however, due to the fact that their dual power
cells are empty in the power diagram for the chosen weights. Interestingly,
there is structure in this family of regular tessellations for S. This becomes
transparent when interpreting such a tessellation, T , in the two different
ways below.
From a combinatorial point of view, T can be associated with the point
q(T ) = W in the high-dimensional configuration space Rn of weights, for
n = |S|. From a geometric point of view, T corresponds to the lower part
of a convex hull in Rd+1 (Theorem 6.3(c)), that is, to a convex simplicial
surface ϕ(T ) in (d + 1)-space. By its convexity, ϕ(T ) minimizes the volume
below it, over all possible (in general, nonconvex) simplicial surfaces based
on the same vertex set in (d + 1)-space.
Volume minimality for ϕ(T ) translates to extremality of q(T ) in the
n-space of weights, meaning that the points q(T ) are the vertices of a convex
polytope, Q(S), in Rn , which is just their convex hull. This polytope of
90
Chapter 6. Higher Dimensions
regular triangulations is called the associahedron in Lee [490] and Gel’fand
et al. [357], and the secondary polytope in Billera et al. [132]. As an
important fact, each edge e = q(T )q(T ) of Q(S) corresponds to a socalled bistellar flip operation between the two regular tessellations T and T for S.
A bistellar flip operates on a subset A of d + 2 weighted sites from S.
There are exactly two ways of partitioning the convex hull, conv(A), of A
into simplices using vertices from A, and the flip replaces
one by the other.
The total number of d-simplices spanned by A is d+2
d+1 = d + 2, and the
flip — depending on its type — deletes k of them and constructs d+2−k new
ones, for some 1 ≤ k ≤ d + 1. For the types k ≤ d, the subset A is in convex
position, whereas for the type k = d+1, one site p ∈ A is interior to conv(A).
In the last case, the flip constructs only a single d-simplex, conv(A), and
leaves p isolated. That is, p is not a vertex of the new simplicial complex any
more, and we talk of a (vertex-)removing flip. (In the dual environment, this
means that the cell of p is now empty in the power diagram for the current
weights.) By contrast, the former flip types may be called exchanging flips,
since they just exchange certain d-simplices but leave unchanged the vertex
set of the simplicial complex.
For d = 3, the exchanging flip of type 2-to-3 (for tetrahedra) is described
and illustrated in Section 6.1.
The existence of the secondary polytope Q(S) has an important
implication. Since the edge graph of a convex polytope is connected, there
exists for any two regular tessellations T1 and T2 of S, a sequence of flips
that takes T1 into T2 .
In fact, a stronger result holds, as has been proven recently in Pournin
and Liebling [595]: T1 and T2 are still connected in the edge graph of Q(S)
if all edges that correspond to removing flips are ignored. This means that
any two regular tessellations of S can be transformed into each other by
using exchanging flips only, that is, without ever removing a vertex.
6.3.3. Flipping and lifting cell complexes
If we leave the class of regular simplicial complexes, flipping becomes
difficult. The main reason behind is that an interpretation in terms of
convex surfaces (Theorem 6.3(c)) is missing.
Even the classical result in R2 , that an arbitrary triangulation of S
can be flipped into the Delaunay triangulation DT(S) by Delaunay flips
(Section 4.2), does not generalize if a specific regular triangulation is to be
reached: We can define locally regular faces in a simplicial complex on S,
similar to locally Delaunay faces — based on the notions of obtuse and acute
6.3. Regular simplicial complexes
91
pairs of weighted sites introduced in the beginning of this section — but
global regularity will not be reached if only regularizing flips are to be
used; see e.g. [30]. This is due to the fact that not every interior edge of a
triangulation might be flippable, see below. Still, the class of triangulations
is connected under arbitrary, in general non-regularizing, edge-exchanging
flips.
In particular, O(n2 ) is an upper bound on the flip distance between
any two triangulations T1 and T2 of a given n-point set S in the plane:
T1 as well as T2 can be flipped into DT(S), and by reversing the latter flip
sequence, T1 transforms into T2 . On the way to DT(S), any fixed edge —
once flipped away — will never reappear, because the corresponding lifted
surface (see Subsection 6.3.2), which now lives in 3-space, will decrease in
height until the convex triangular surface ϕ(DT(S)) is attained. As n2
edges are spanned by S, the above bound follows. It is asymptotically
tight, as an Ω(n2 ) lower bound exists; see e.g. Fortune [343]. The problem
of computing a shortest flip sequence that transforms two triangulations
into each other has recently been shown to be NP-hard in Lubiw and
Pathak [513] and Pilz [591].
We observe that not all edges in a triangulation of S can be flipped.
This is trivially true for each convex hull edge e, but also if e borders
two triangles whose union is nonconvex. Hurtado et al. [411] have shown
flippable edges. When
that every triangulation contains at least (n−4)
2
transforming triangulations, these can be flipped in parallel, as long as
they are independent, i.e., they do not share a vertex (or equivalently, they
do not belong to the same triangle). In fact, linearly many edges always are
both flippable and independent, leading to a flip distance of only O(n) for
this more powerful parallel operation; see Galtier et al. [348], and recently
which is tight.
Souvaine et al. [661] for an improved bound of (n−4)
5
In 3 dimensions already, it is an open problem whether every
tetrahedral complex can be turned into DT(S) by a sequence of exchanging
flips. An affirmative answer would imply that any two given tetrahedral
complexes of a finite point set in R3 can be flipped into each other. In fact,
for dimensions 5 (and higher), a counterexample has been found recently
by Santos [616]. On the other hand, flipping works in general dimensions
for the incremental construction algorithm in Edelsbrunner and Shah [313],
because there site insertions take place already in a regular tessellation for
the sites processed so far, which guarantees a special structure.
The practical relevance of the flipping problem above is given by a
result in Guibas and Russel [393]. They have shown by experiments that
flipping will convert to DT(S) certain ‘near’-Delaunay tetrahedrizations
(stemming from small vertex movements, for example) up to three times
92
Chapter 6. Higher Dimensions
Figure 6.5. Pseudo-triangulation with 13 vertices. A pseudo-triangle with many vertices
is highlighted.
faster than DT(S) can be constructed from scratch, provided flipping does
not get stuck.
Flippability can be retained if we extend the class of simplicial
complexes to so-called pseudo-simplicial complexes.
Of particular interest are pseudo-triangulations in the plane, which
contain, in addition to triangles, general polygonal faces with exactly
three convex interior angles. See Figure 6.5 for an illustration. Pseudotriangulations have many properties useful in diverse applications and are
investigated, concerning their flipping properties and embedding properties
into polygonal surfaces, in Aichholzer et al. [30]. Flip distances of O(n log n)
between arbitrary triangulations are achieved, using ‘shortcuts’ via
(less edge-consuming hence more flexible) pseudo-triangulations. Pseudosimplicial complexes in general dimensions are introduced and studied in
Aurenhammer and Krasser [101]. Both papers also give short summaries
of earlier literature on pseudo-triangulations; a comprehensive survey is
provided in Rote et al. [609]. An even further relaxation of triangulations,
called pre-triangulations in Aichholzer et al. [33], allows the definition and
characterization of various polygonal partitions with special flipping and
lifting properties.
In fact, deciding the liftability to 3-space of a given polygonal partition
of the plane is an intriguing topic. Various geometric and combinatorial
results exist, see e.g. [30, 33] and references given there. A classical
result is known as Steinitz’s theorem [663], presented nicely in Ziegler’s
textbook [717]: Every triangulation in the plane, as being a 3-connected
planar graph, is combinatorially equivalent to the edge graph of some
convex polyhedron in 3-space.
6.4. Partitioning theorems
93
Finally, let us consider so-called constrained simplicial complexes, which
are restricted to the interior of nonconvex polytopes. A most popular
representative we have encountered in Section 5.4 is the constrained
Delaunay triangulation of a polygon (or, more generally, of a given set of line
segments). While every polygon can be triangulated, constrained regular
triangulations either do not always exist, or are not unique in general,
depending on their definition; see [30]. In dimensions higher than two,
another difficulty arises: Not every nonconvex polytope admits a simplicial
complex without extraneous vertices. The Schönhard polytope [621, 717]
is a well-known counterexample in R3 . In fact, it is NP-hard to decide
whether a nonconvex polytope in R3 can be tetrahedrized; see Ruppert
and Seidel [613].
Using a different construction, Grislain and Shewchuk [381] showed
that deciding the existence of a constrained Delaunay tetrahedrization
in R3 remains NP-hard if co-spherical degeneracies in the vertex set of the
underlying polytope are allowed, and that the problem has a polynomial
solution in the non-degenerate case.
A thorough treatment of constrained Delaunay (and regular)
tessellations in general dimensions is given in Shewchuk [651]. For example,
a sufficient existence condition is that each constraining (d − 2)-simplex
is separable by a sphere from the remaining vertices in the tessellation.
Shewchuk’s papers also describe implementations and applications of
constrained Delaunay and regular simplicial complexes, in the fields of mesh
generation and finite element methods.
When generalizing to pseudo-simplicial complexes, unique existence and
flippability of constrained regular tessellations is guaranteed; see [30, 33].
This is due to the existence of respective convex polytope representations
in the high-dimensional configuration space Rn , which constitute generalizations of the secondary polytope discussed in Subsection 6.3.2.
6.4. Partitioning theorems
Voronoi diagrams partition their underlying space according to a given
set of sites and a distance function. In this sense, they naturally induce a
clustering of space (or of possible objects situated therein). We will discuss
this fact now, in the context of power diagrams, because the possibility
of weighting the sites gives room for interesting existence and optimality
results; see Aurenhammer et al. [97]. The general role of Voronoi diagrams
in geometric clustering is described in Section 8.4.
94
Chapter 6. Higher Dimensions
6.4.1. Least-squares clustering
Consider a set S of n point sites, and another set X of m points, in
d-space Rd . The (closest-site Euclidean) Voronoi diagram of S defines an
assignment function A : X → S, given by
A(x) = s
if and only if x ∈ VR(s).
Here VR(s) denotes the Voronoi region of site s ∈ S. Equivalently, A−1 (s) =
X ∩ VR(s), for all s ∈ S. (Points of X that have more than one closest site
in S are not uniquely covered by this definition; by convention, A assigns
each such point to an arbitrary but fixed closest site.) The total number
of points assigned to a particular site s, |A−1 (s)|, is called the capacity
of s. The capacities of all sites add up to m = |X|. The assignment A has
an obvious optimization property: It minimizes the sum of the distances
between sites and their assigned points, over all possible assignments of X
to S.
Given S and X, we would like to be able to change the assignment by
varying the distance function that underlies the Voronoi diagram of S. To
this end, we attach a set W = {w(s) | s ∈ S} of weights to the sites, and
replace the Euclidean distance d(x, s) between a point x and a site s by the
power function
powW (x, s) = d(x, s)2 − w(s).
In the resulting power diagram (see Section 6.2), each region is still a convex
polyhedron, and has the property of shrinking (or expanding) when the
weight of its defining site is decreased (respectively, increased). As above,
we obtain an assignment function AW : X → S which now depends on the
choice of weights. In particular, the site capacities depend on W .
A least-squares clustering of a set X of points with respect to a
set S of sites (or cluster centers) in d-space is defined to minimize the
total squared Euclidean distance between the sites and their associated
points. Our interest is in constrained least-squares clusterings, where the
number of associated points per site (i.e., its capacity) is prescribed.
Interestingly, assignments defined by power diagrams are constrained leastsquares clusterings.
Lemma 6.1. Let S and X be finite sets of sites and points in Rd ,
respectively, and fix a set W of weights for S. The assignment AW
minimizes
d(x, A(x))2
x∈X
6.4. Partitioning theorems
95
over all assignments A : X → S with capacity constraints |A−1 (s)| =
|A−1
W (s)| for all s ∈ S.
Proof. From the definition of AW it is evident that AW minimizes the
expression
powW (x, A(x)) =
d(x, A(x))2 −
w(A(x))
x∈X
x∈X
x∈X
over all possible assignments A : X → S, regardless of the capacity
constraints. The last sum, being equal to s ∈ S |A−1 (s)| · w(s), is a fixed
constant for all assignments A with capacities |A−1 (s)| = |A−1
W (s)|. The
lemma follows.
Power diagrams also give rise, in the obvious way, to mappings of the
entire d-space to the set of sites. Let AW : Rd → S be the assignment
induced by the power diagram of S with weights W . That is, A−1
W (s) =
regW (s) is the power cell of site s in the diagram. The capacity of a site
now can be defined, for example, as the fraction of the unit hypercube [0, 1]d
contained in its power cell.
Formally, let be a continuous and non-vanishing probability
distribution on [0, 1]d, and let µ(X) = X (x)dx denote the measure of
a set X ⊂ Rd with respect to . Then µ(A−1
W (s)) is the capacity of s that
results from AW . The capacities of all sites add up to 1. A continuous
version of Lemma 6.1 can be proved easily. In fact, we have the following
general result [97].
Theorem 6.4. Let S be a finite set of sites in Rd . Any (discrete or
continuous) assignment induced by a power diagram of S is a least-squares
clustering, subject to the resulting capacities. Conversely, a least-squares
clustering for S, subject to any given capacities (whose sum is the total
number of assigned points in the discrete case, and 1 in the continuous
case) exists and can be realized by a power diagram of S.
For finite point sets X, the existence of a constrained least-squares
clustering L : X → S is trivial. Its realizability by power diagrams
follows from an algorithm that constructs such a power diagram, given
in Subsection 6.4.2. Existence and realizability in the continuous case,
L : [0, 1]d → S, can be proved by using the fact that, for any two
sites s, t ∈ S, the sets L−1 (s) and L−1 (t) are separable by a hyperplane.
An alternative proof is given in Cuesta-Albertos and Tuero-Diaz [236],
in the more general context of the so-called Monge–Kantorovich mass
transference problem; see Gangbo and McCann [350], Caffarelli et al. [175],
and Subsection 7.4.1.
96
Chapter 6. Higher Dimensions
The existence of least-squares clusterings, together with their
realizability by power diagrams, immediately implies several partitioning
results for power diagrams. The discrete variant reads as follows.
Theorem 6.5. Let S and X be a set of n sites and m points in Euclidean
d-space Rd , respectively. For any choice of integer site capacities c(s) with
−1
s ∈ S c(s) = m, there exists a set W of weights such that |AW (s)| = c(s),
for all sites s ∈ S.
In other words, there always exists a power diagram whose regions
partition a given d-dimensional finite point set X into clusters of prescribed
sizes, no matter where the sites of the power diagram are chosen. Moreover,
we have the following continuous version of Theorem 6.5.
Theorem 6.6. Let S be a set of n sites in Rd , let be a continuous
and non-vanishing probability distribution on [0, 1]d , and let µ denote the
measure with respect to . For any capacity function c : S → [0, 1] satisfying
s ∈ S c(s) = 1, there is a set W of weights such that µ(regW (s)) = c(s),
for all sites s ∈ S.
By taking, for instance, to be the uniform distribution in [0, 1]d we
get the following corollary.
Corollary 6.1. For any set of n sites in Rd there exists a power diagram
that partitions the unit hypercube into n polyhedral regions of prescribed
volumes.
This seems surprising, as the placement of the sites determines the
normals of the facets separating the power regions. Corollary 6.1 is, in a
close sense, related to Minkowski’s theorem for convex polytopes (see e.g.,
Grünbaum [384]): Let V be any collection of n non-zero non-parallel vectors
that span Rd+1 and sum up to zero. Then there exists a convex (d + 1)polytope with n facets in one-to-one correspondence with vectors of V , such
that each facet is normal to its corresponding vector and has d-dimensional
volume equal to the vector length. Though each power diagram is a
projection of a convex polytope (Section 6.2), the statements are not
equivalent, however, due to the presence of unbounded components in the
latter polytope. Similar is also the partitioning result in Aurenhammer [94]
for convex polygons in the plane, where the so-called weighted skeleton of a
polygon (a weighted variant of the straight skeleton discussed in Section 5.3)
is used to optimally partition various sets of objects.
A recent result on equipartitions into convex sets (i.e., partitions
into sets of equal volume), which takes surface areas into account, is
6.4. Partitioning theorems
97
based on power diagrams as well; see Hubard and Aronov [409]. For
further partitioning results, applications, and references; see e.g., Carlsson
et al. [184], and Subsection 7.4.1.
6.4.2. Two algorithms
It is of practical interest to compute the partitions in Theorems 6.5 and 6.6
efficiently. In the discrete case, the following point insertion algorithm can
be used. The continuous case can be treated with a gradient method,
described later in this subsection.
The incremental insertion algorithm computes, for a set S of n sites
and a set X of m points in Rd , a least-squares clustering L : X → S subject
to a given integer capacity vector c with s ∈ S c(s) = m. By Lemma 6.1,
it suffices to compute a weight vector W = (w(s))s ∈ S such that we have
|X ∩ regW (s)| = c(s) for all s ∈ S.
We start with W = 0, for which the power diagram is just the classical
Voronoi diagram of S, and proceed in m phases, each inserting one point x ∈
X into the current diagram. The vector W , and with it the power diagram,
is then recomputed such that the invariant b(s) ≤ c(s) is maintained for all
s ∈ S, where b(s) denotes the current number of points in regW (s).
(1) Add point x to the region regW (s) containing x in the current power
diagram. If now b(s) ≤ c(s) the phase ends, as there is no need to
change W . Otherwise, let D = {s}. Intuitively, the set D will contain
the sites whose regions are too large and must be shrunk.
(2) Repeat the following two steps (consult Figure 6.6):
(a) Shrink all D-regions by simultaneously decreasing their weights.
More formally, find the smallest positive number ∆ so that
decreasing the weights of all D-sites simultaneously by more than
∆ causes one of the shrinking regions to lose a point, say p . Notice
that in this process a site in S \ D cannot lose a point to a D-site,
and that no point can move between two D-regions or between two
non-D-regions.
(b) Decrease the weights of all D-sites by ∆. Consider the region reg(s )
where p would end up, had we shrunk the weights by more than ∆.
If b(s ) < c(s ) then go to Step 3, as we have found a region which
is not full. Else add s to D and repeat (a).
(3) We have found a region reg(s ) that is not full and a point p on its
boundary. Assign p to s . This makes some region reg(s ) with s ∈ D
less than full. But s was added to D because of some point p that it
shared with a site s that had already been in D. So assign p to s
98
Chapter 6. Higher Dimensions
x
Figure 6.6. A set of D-regions (shaded) losing a point. After inserting point x in the
lower left corner, reassignment takes place as indicated by arrows.
and follow the chain back, reassigning points to regions and adjusting
b-values accordingly, until the original site s is encountered and relieved
of one point.
This procedure can be implemented in O(n2 m log m + nm log2 m) time
and O(m) space, for the (probably most interesting) planar case [97]. This
improves over the O(nm2 +m2 log m) time and O(nm) space algorithm that
results from transforming the problem into a minimum-cost flow problem;
see Fredman and Tarjan [345]. Note that, in the terminology of network
flows, the chain-like process of reassigning points to sites at the end of a
phase corresponds exactly to an augmenting path.
For the continuous problem, a gradient method for iteratively improving
the weight vector W can be used. In fact, for a fixed set of sites, the value of
the assignment induced by the power diagram with weights W is a concave
function of W . Finding a weight vector such that the resulting assignment
fulfills the capacity requirements is then equivalent to finding the maximum
of a related function whose domain is the configuration space of weights.
Let be a continuous and non-vanishing probability distribution in
[0, 1]d . For an arbitrary but fixed assignment A : [0, 1]d → S, define the
function fA : Rn → R by
fA (W ) =
[0,1]d
(x) · powW (x, A(x))dx.
6.4. Partitioning theorems
99
Let B(A) = (µ(A−1 (s)))s ∈ S be the vector of capacities resulting from the
assignment A, and put
(x) · δ(x, A(x))2 dx,
Q(A) =
[0,1]d
the value of A. With this notation, fA can be written as
fA (W ) = −B(A) · W + Q(A).
Hence fA is a linear function of W . Now consider the function f : W →
fAW (W ); recall that AW is the assignment induced by the power diagram
with weights W . We claim that f is the pointwise minimum of the class
of functions fA . Indeed, for fixed W , the assignment AW minimizes the
value fA (W ) by definition of the power diagram of S with weights W . In
other words, the graph of f is the lower envelope of an (infinite) set of
hyperplanes in Rn+1 , showing that f is a concave function.
By the choice of properties of , B(AW ) and Q(AW ) depend continuously on W . Hence, for each W = W , the graph of f has at point
(W , f (W )) a unique tangent hyperplane xn+1 = −B(AW ) · W + Q(AW )
that changes continuously with W . That is, f describes a smooth surface.
Note that the gradient f (W ) of f at W is given by −B(AW ).
Recall that we aim to find a weight vector W ∗ such that B(AW ∗ ) = C,
the given capacity vector. Consider the function
g(W ) = f (W ) + C · W
= (C − B(AW )) · W + Q(AW ).
Its gradient g(W ) is C − B(AW ), hence our requirement B(AW ∗ ) = C
just means g(W ∗ ) = 0. This corresponds to a global maximum of the
smooth concave function g. So the problem we want to solve is: Find W ∗
such that g(W ∗ ) is maximized.
Finding the maximum of a concave and smooth n-variate function
is a well-studied problem. In our case, we can exploit the fact that, for
any given weight vector W , we can compute g(W ) and g(W ) from the
power diagram with weights W . So a gradient method (see, e.g., [129]) for
iteratively approximating W ∗ can be used. Starting, for example, with the
weight vector W0 = 0 (corresponding to the Voronoi diagram of S), we use
the iteration scheme
Wk+1 = Wk + tk g(Wk ).
If the step sizes tk are chosen properly then Wk converges to the solution
W ∗ at a superlinear rate. Intuitively, what happens is that the weights of
100
Chapter 6. Higher Dimensions
those sites whose region measures are too small (too large) are increased
(respectively, decreased) at each step.
If S is a set of n sites in the plane, and is the uniform distribution in
the square, each step can be carried out in O(n log n) time: For the current
weight vector Wk , we need O(n log n) time to construct the power diagram
of S and Wk , and time O(n) is needed in addition to calculate the area and
the integral of squared distances for each region within the unit square. The
space requirement is optimal, O(n).
6.4.3. More applications
We conclude this section with some applications of the discrete least-squares
clustering problem. As before, let S and X be finite sets of n sites and
m points in Rd , respectively.
For Y ⊂ X and s ∈ S, define the variance of the cluster Y with respect
to the site s as x ∈ Y δ(x, s)2 . Then a constrained least-squares clustering
L : X → S is just a clustering for X such that the clusters have prescribed
sizes and the sum of cluster variances is minimized. Besides being optimum
in the above sense, these clusters have the important property that their
convex hulls are pairwise disjoint: Distinct clusters are contained in different
regions of a power diagram, and power regions are convex. Hull-disjointness
is a natural and desirable property of clusters which, for instance, eases the
classification of new points. Simple examples show that replacing variance
by the sum of distances destroys hull-disjointness.
If we define the profit of cluster Y with respect to site s as x ∈ Y x · s,
then L maximizes the sum of cluster profits for given cluster sizes: We
have δ(x, s)2 = x2 + s2 − 2x · s, and the sum of the first two terms is
independent of the assignment, provided capacity constraints are satisfied.
This concept is motivated by the following transportation problem: Interpret
a point x = (x1 , . . . , xd ) as a truck loaded with xi units of the ith good,
and a site s = (s1 , . . . , sd ) as a market that sells the ith good at price si
per unit. Choose the site capacities according to the attractiveness of the
markets, and L will tell you where each truck should go in order to achieve
maximal profit for these capacities.
The next application makes use of the property that constrained leastsquares clusterings are invariant under translation and scaling.
Observation 6.1. Let σ > 0 and τ ∈ Rd , and consider a least-squares
clustering L : X → S with capacities c. Then σL + τ is a least-squares
clustering of X to σS + τ subject to c.
6.4. Partitioning theorems
101
Consider the special case that S and X are of equal cardinality n, and
let L : X → S be a least-squares clustering subject to c(s) = 1 for all s ∈ S.
L is called a least-squares matching in this case. Define a (one-to-one) leastsquares fitting as the least-squares matching L∗ : X → σS + τ where the
value of L∗ is minimal over all positive scaling factors σ and all translation
−1
(s) for all s ∈ S.
vectors τ . Observation 6.1 tells us that L−1
∗ (σs + τ ) = L
Thus, when computing the least-squares fitting L∗ , we can first calculate
and fix the matching L, as a least-squares matching of X to S, and then
determine the optimizing values of σ and τ for this matching, which is an
easy analytic task.
In this way, a least-squares fitting of size n in R2 can be computed in
3
O(n ) time and O(n) space, and a least-squares fitting of m points to n sites
subject to given capacities can be computed in O(n2 m log m + nm log2 m)
time and O(m) space; see [97] for details. This improves the best known
least-squares matching algorithm (a variant of Vaidya’s algorithm [689]),
with O(m2+ε ) time and O(m1+ε ) space, for m > n2 .
Recently, partial least-squares matchings in R2 have been studied in
Rote [608]. We are given some point set X (the ‘pattern’) of size m, where
m is now smaller than the size n of the set S of sites (the ‘ground set’).
Consider the assignment A : X → S that minimizes
d(x, A(x))2
x∈X
for site capacities which are not fully prescribed but only constrained
to c(s) ∈ {0, 1}. Unlike the fully constrained case (covered in Lemma 6.1),
the assignment A now depends on the geometric position of X. Site
capacities will be 1 or 0 as demanded by optimality. Note that A is an
injective mapping. In view of an efficient pattern matching algorithm,
the following question arises: How many different optimal assignments are
there when the pattern X is translated in the plane? As a partial answer,
m(n − m) + 1 is a tight bound when X is moved along a straight line. This
implies a polynomial-time algorithm for this restricted case.
If X is moved but slightly, the assignment will most likely remain
unchanged. Now a partition of the plane into cells can be defined, such that
each cell reflects all translations of X with the same optimal assignment.
This partition is just the classical Voronoi diagram of S if X is a singleton
set (i.e., m = 1), and its cells are convex polygons for general m; see
Figure 6.7 for the case m = 6.
Henze et al. [403] showed that the size of this partial-matching partition
is O(m! m2 n4 ) (and polynomial in n for any fixed dimension d). The
conjectured complexity is O(m2 n2 ), as this is also the complexity of the
Chapter 6. Higher Dimensions
102
Figure 6.7. Partial-matching partition of the plane. A pattern (×) of size 6 is matched
to a ground set of 18 sites (•); from [608].
overlay of m Voronoi diagrams, each for n sites. The overlay of two convex
partitions C1 and C2 of the plane is the (convex) partition obtained by
superimposing C1 and C2 ; new vertices are created by edges that cross,
hence new edges and polygons arise as well.
We mention that every partial-matching partition is actually a power
diagram, as a convex projection polyhedron in 3-space can be defined.
Another related assignment problem has been investigated in Brieden
and Gritzmann [167]. In a balanced fractional clustering, each of the m
points x ∈ X is allotted an ‘area’ a(x), and each of the n sites (cluster
centers) s ∈ S has a given capacity c(s). Here a(x) and c(s) are real-valued,
and m ≥ n. (In a more relaxed variant, some small interval [c− (s), c+ (s)]
of valid capacities per site may be specified instead.) Areas and capacities
obey the condition
a(x) =
c(s).
x∈X
s∈S
Now, fractions ξ1 (x), . . . , ξn (x) of each point x, with
ξi (x) = 1, are to be
assigned to the sites s1 , . . . , sn such that
ξi (x)a(x) = c(si ) (or ∈ [c− (si ), c+ (si )]) for all i.
x∈X
This clustering problem is motivated by an application in farmland
consolidation. The n farmers residing at sites s1 , . . . , sn cultivate a total
6.5. Higher-order Voronoi diagrams
103
of m lots x ∈ X with individual areas a(x), and they want to exchange lots
so as to move them closer together for efficiency purposes. Of course, the
original farm sizes c(si ) should (almost) stay the same by reassignment.
In its integer version, deciding the existence of a balanced fractional
clustering is NP-hard; see Borgwardt et al. [154].
As is shown in [167], certain types of balanced fractional clusterings are
induced by power diagrams of the sites in S, and in particular, by so-called
centroidal power diagrams, where the sites coincide with weighted centroids
of their clusters; cf. Subsection 9.3.4.
6.5. Higher-order Voronoi diagrams
Higher-order Voronoi diagrams are natural and useful generalizations of
the classical Voronoi diagram. Given a set S of n point sites in d-space Rd
and an integer k between 1 and n− 1, the order-k Voronoi diagram of S, for
short Vk (S), partitions the space into regions such that each point within
a fixed region has the same k closest sites. V1 (S) just is the (closest-site)
Voronoi diagram V (S) of S.
Many of the applications of the classical Voronoi diagram (in Chapter 8
and other chapters) are meaningful in their ‘order-k’ versions as well,
like k-nearest neighbor search and largest ‘almost’ empty circle placement;
or higher-order Voronoi diagrams apply directly, like for clustering and
classification problems (Section 8.4).
The regions of Vk (S) are convex polyhedra, as they arise as the
intersection of halfspaces of Rd bounded by symmetry hyperplanes of the
sites. A subset M of k sites in S has a non-empty region in Vk (S) iff there
is a sphere that encloses M but no site in the complement S \ M . In fact,
the region of M in Vk (S) just is the set of centers of all such spheres.
Figure 6.8 illustrates a planar order-2 Voronoi diagram. Two differences
between this and the classical Voronoi diagram are apparent. A region need
not contain its defining sites, and the bisector of two sites may contribute
more than one edge (or facet, in higher dimensions) to the diagram.
6.5.1. Farthest-site diagram
For the largest admissible value k = n − 1, a subset M of S of size n − 1 has
to be closest to a point x ∈ Rd in order to claim x for its region. In other
words, the singleton site {p} = S \ M has to be farthest from x. For this
reason, the diagram Vn−1 (S) is commonly called the farthest-site Voronoi
diagram of S. It contains, for each site p ∈ S, the region of all points x for
which p is the farthest site in S.
Chapter 6. Higher Dimensions
104
q
p
Figure 6.8.
Region of {p, q} in the order-2 Voronoi diagram V2 (S).
This diagram has special structural properties. For example, only sites
lying on the boundary of the convex hull of S, and exactly those, have
non-empty regions in Vn−1 (S). This is because a site interior to the convex
hull can never be the farthest from any point x in Rd . Moreover, all regions
are unbounded, as each of them contains an unbounded part of some ray
emanating from the defining site and pointing ‘into’ S. In the plane, this
implies that the edge graph of Vn−1 (S) is a tree. It consists of exactly 2h−3
edges and h−2 vertices, if h denotes the number of extreme points in S (and
general position is assumed; the size of Vn−1 (S) is less, otherwise). For an
illustration, see Figure 6.10 in Subsection 6.5.4, where several farthest-site
Voronoi diagrams are drawn in dashed style.
Exact upper bounds on the size of farthest-site Voronoi diagrams in Rd
for d ≥ 3 are derived in Seidel [628].
Several methods of construction apply to Vn−1 (S). Let us first consider
the lifting approach taken in Subsection 6.2.2, where distances to the sites
pi ∈ S are described by hyperplanes π(pi ) in Rd+1 . Clearly, the weights
w(pi ) have all to be put to 0 now, as we deal with Euclidean distances
rather than with power functions.
6.5. Higher-order Voronoi diagrams
105
As farthest distances are to be used, the lower envelope of the
hyperplanes π(p1 ), . . . , π(pn ) will vertically project to Vn−1 (S). Consequently, by duality this implies that the upper part of the convex hull of
their set of poles in Rd+1 , {λ(pi ) | pi ∈ S}, corresponds to the farthestsite Delaunay complex, DTn−1 (S), of S. This simplicial complex includes a
d-simplex spanned by d+1 sites in S if and only if its circumsphere encloses
the entire set S. Observe that such simplices are exclusively spanned by
extreme points of S, such that no vertices of the complex DTn−1 (S) are
interior to the convex hull conv(S) of S in Rd .
For example, in the planar case, DTn−1 (S) is just a triangulation of
the vertices of the convex polygon conv(S). We called it the farthest-site
Delaunay triangulation. This structure has an optimality property, as was
shown by Eppstein [321]: It minimizes angularity, and thus, the smallest
occurring angle, among all possible triangulations for the convex hull points.
By contrast, the Delaunay triangulation DT(S) maximizes angularity; see
Section 4.2.
Algorithmically, the discussion above implies that both structures
Vn−1 (S) and DTn−1 (S) come for free, when we compute a convex hull
in Rd+1 in order to obtain the classical (closest-site) Voronoi diagram
V (S) = V1 (S) or the Delaunay complex DT(S) = DT1 (S) in Rd , as done
in Sections 3.5 and 6.2.
When we are not interested in closest-site structures, the farthest-site
Delaunay triangulation in R2 is preferably computed directly, with a simple
ear-clipping algorithm which also achieves an O(n log n) runtime, and which
we shall sketch below.
Let C be a convex polygon. An ear of C is a triangle which shares two
edges with C. Each triangulation T of C (and of any simple polygon) has
at least two ears, as these correspond to leaves in the dual tree of T . The
algorithm now constructs the h − 2 triangles building DTn−1 (S) ear by ear.
Recall that h ≤ n denotes the number of extreme points of S, that is, the
number of vertices of the polygon conv(S).
Initially, by scanning the boundary of conv(S), one of its ears, ∆, is
identified whose circumcircle encloses S. The largest of the h encountered
circles will have this property. The ear ∆ is clipped from conv(S), and the
two newly created ears adjacent to ∆ are examined, and their circumradii
put into a priority queue Q, which initially holds the h circumradii checked
before. In this way, at any time the currently largest radius stored in Q
gives us the next ear to be clipped off, and with it, one more triangle of
DTn−1 (S). It is clear that only O(h) insertions and maxima deletions take
place in Q. Either operation can be performed in time O(log h). So, the
O(n log n) time spent for computing conv(S) dominates the runtime. The
106
Chapter 6. Higher Dimensions
diagram Vn−1 (S), if desired, can be derived from DTn−1 (S) in additional
O(h) time.
A similar ear-clipping algorithm is used in Devillers [262], for
re-triangulating the ‘hole’ that results from removing a site p and its
incident faces from the (classical) Delaunay triangulation DT(S). (Rather
than being convex, this hole is a star-shaped object; all its parts are visible
from p.) He also bases his clipping sequence on some shelling order of the
triangles, in DT(S \ {p}). This order can be found by considering planes
through the projections of the ear triangles onto a vertical paraboloid in
3-space (see Section 3.5), because these planes intersect the rotation axis
of the paraboloid in some shelling order of the Delaunay triangles. The
method works in general dimensions d, and constitutes a corrected version
of Heller’s planar algorithm [401]. Shelling orders of convex polytopes play
a role in the computation of higher-dimensional convex hulls (and thus, of
Delaunay, Voronoi, and power complexes); see e.g. Seidel [633].
Farthest-site Voronoi diagrams are meaningful for distance functions
other than the Euclidean, and for sites more general than points. In fact,
the planar shelling algorithm described above generalizes within the same
runtime bound, O(n log n), to the farthest-line segment Voronoi diagram;
see Aurenhammer et al. [95] and Section 5.1.
We remark that the maximal size of generalized farthest-site Voronoi
diagrams may be drastically smaller than that of their closest-site counterparts. This is briefly discussed at the beginning of Subsection 7.4.3.
6.5.2. Hyperplane arrangements and k-sets
The family of all occurring higher-order Voronoi diagrams for a given set S
of point sites in d-space Rd , V1 (S), V2 (S), . . . , Vn−1 (S), is closely related to
an arrangement of hyperplanes in Rd+1 . This fact has been observed in the
seminal paper by Edelsbrunner, O’Rourke, and Seidel [309]. We describe
this relationship in the more general setting of power diagrams, by defining
an order-k power diagram, PDk (S), for a set S of weighted point sites in
an analogous way; see Aurenhammer [87, 93].
Recall from Section 6.2 that the power function, pow(x, p), with respect
to a site p in Rd can be expressed by a hyperplane π(p) in Rd+1 .
The set of hyperplanes {π(p) | p ∈ S} dissects (d + 1)-space into a
polyhedral cell complex called a hyperplane arrangement. Arrangement
cells are convex, and can be classified according to their relative position
with respect to the hyperplanes in {π(p) | p ∈ S}. A cell C is said to be of
level k if exactly k hyperplanes are vertically above C. For example, the
upper envelope of {π(p) | p ∈ S} (i.e., the pointwise maximum of these linear
6.5. Higher-order Voronoi diagrams
107
Π (S)
x’
Figure 6.9. Arrangement of six lines in R2 with level-2 cells shaded. Each point x in a
fixed level-2 cell projects vertically to a particular region of the respective order-2 power
diagram.
functions) bounds the only cell, Π(S), of level 0. All cells of level 1 share
some facet with Π(S), so that their vertical projection gives the (order-1)
power diagram PD(S).
More generally, the cells of level k project to the regions of PDk (S),
for each k between 1 and n − 1. To see this, let x ∈ Rd+1 be a point in
some k-level cell C. Then k hyperplanes π(p1 ), . . . , π(pk ) are above x , and
n − k hyperplanes π(pk+1 ), . . . , π(pn ) are below x . That means that, for
the vertical projection x of x onto Rd , we have pow(x, pi ) < pow(x, pj ) for
1 ≤ i ≤ k and k + 1 ≤ j ≤ n. Hence x is a point in the region of {p1 , . . . , pk }
in PDk (S). Figure 6.9 gives an illustration for d + 1 = 2.
Hyperplane arrangements are well-investigated geometric objects,
concerning their combinatorial as well as their algorithmic complexity;
see e.g. Edelsbrunner et al. [309, 311], and especially the monograph by
Edelsbrunner [300]. Their size does not vary with the particular placement
of the hyperplanes, and an optimal insertion-based construction algorithm
in general dimensions exists. From the discussion above, we obtain the
following theorem.
Theorem 6.7. Let S be a set of n (weighted or unweighted) point sites
in Rd . The family of all higher-order power diagrams (or Voronoi diagrams)
for S realizes a total of Θ(nd+1 ) faces, and it can be computed in optimal
Θ(nd+1 ) time.
108
Chapter 6. Higher Dimensions
Clarkson and Shor [229] proved that the total number of cells in
an hyperplane arrangement in Rd , with levels not exceeding a fixed
value of k, is bounded by O(nd/2 k d/2 ). The collection of these cells
can be constructed within this time for d ≥ 4, with the algorithm in
Mulmuley [554]. A modification by Agarwal et al. [10] achieves roughly
O(nk 2 ) time in R3 . An output-sensitive construction algorithm is given in
Mulmuley [555].
Bounding the combinatorial complexity of a single k-level (for a fixed
value of k) is an intriguing problem. The maximal size of a k-level, over
all possible arrangements of n hyperplanes in Rd , increases with k ≤ n2 ,
though not necessarily in a monotonic way for a given set of hyperplanes. To
mention a few results, Barany et al. [112], and Dey and Edelsbrunner [270],
respectively, proved upper and lower bounds of O(n8/3 ) and Ω(n2 log n)
for R3 . Agarwal et al. [9] gave a corresponding upper bound dependent
on k, O(nk 5/3 ), which was improved to O(nk 3/2 ) in Sharir et al. [643].
The
√
currently best known lower bound in Rd , for d ≥ 3, is Ω(nd−1 · 2c log n );
see Sharir et al. [643] and Nivasch [566].
All the results above apply to order-k power diagrams in one dimension
lower. (Note that, interestingly, a high-order power diagram has a larger
worst-case size than a high-order Voronoi diagram; cf. Subsection 6.5.3.)
Moreover, these results also apply to so-called k-sets of a set S of n
points in the same dimension, by the polarity transform between points
and hyperplanes described in Section 6.2. A k-set is a subset of k points
of S that can be separated from S by a hyperplane. In fact, k-sets have
been objects of long-lasting investigation in combinatorial geometry, as is
documented in Goodman and O’Rourke’s handbook [371].
Polarity preserves the relative position between points and hyperplanes.
With this, we observe that each point x in a fixed k-level cell will
dualize into some hyperplane h(x ) that cuts off a unique k-set from the
corresponding set S of poles (the dualized arrangement hyperplanes),
because h(x ) has the same k points of S below it, as long as x stays
inside the same cell.
As each dividing hyperplane splits S into two subsets of cardinalities
k and n − k respectively, for some value of k, the number of k-sets of
S trivially equals the number of its (n − k)-sets. Now observe that each
k-set M of S bijectively corresponds to an unbounded region in the power
diagram PDk (S ) (in the same dimension), because points x far from S but
still closest to M have to exist. We conclude that the number of unbounded
regions in an order-k power (or Voronoi) diagram is exactly the same as
in its order-(n − k) counterpart — a fact not true for non-point sites like
6.5. Higher-order Voronoi diagrams
109
e.g. line segments [584]. Moreover, this number is, in the worst case, of the
same order of magnitude as the number of all regions in an order-k power
diagram in one dimension less.
Similar concepts are k-facets, which are oriented (d − 1)-simplices
spanned by d points in an n-point set S in Rd , such that there are exactly
k points of S in the positive open halfspace determined by such a simplex.
The numbers of k-sets and k-facets lie within constant factors from each
other. In dimension 3, the exact total number of k-triangles, for 1 ≤ k ≤ m,
has been determined in Aichholzer et al. [42] for the range m < n4 .
6.5.3. Computing a single diagram
Many practical applications ask for the computation of a single order-k
Voronoi diagram Vk (S) in the plane, for a fixed value of k. (Typically, k
does not depend on |S| = n but is a small constant.) Like in the classical
Voronoi diagram V (S), edges are pieces of perpendicular bisectors of sites.
Vertices are centers of circles that pass through three sites. However, these
circles are no longer empty; they enclose either k − 1 or k − 2 sites. Lee [495]
showed that this diagram has O(k(n − k)) regions, edges, and vertices. It
is easy to see that the regions of the order-2 Voronoi diagram V2 (S) are in
one-to-one correspondence with the edges of V (S). Hence V2 (S) realizes at
most 3n − 6 regions in the plane.
Considerable efforts have been made to compute the single planar
order-k Voronoi diagram efficiently. Different approaches have been
taken in Lee [495], Chazelle and Edelsbrunner [197], Aurenhammer [92],
Clarkson [224], Agarwal et al. [10], and Ramos [600]. In the last three
papers, randomized runtimes of O(kn1+ε ), respectively O(kn log n +
∗
n log3 n) and O(kn2c log k + n log n) are achieved, which is close to optimal.
For the L1 - and L∞ -metrics, an output-sensitive algorithm has been given
in Liu et al. [509].
Below we describe a (roughly) O(k 2 (n − k) log n) time randomized
incremental algorithm by Aurenhammer and Schwarzkopf [102], that can
be modified to handle arbitrary on-line sequences of site insertions, site
deletions, and k-nearest neighbor queries. Though not being most time
efficient, the algorithm profits from its simplicity and flexibility.
The heart of the algorithm is a duality transform that relates the
diagram Vk (S) to a certain convex hull in 3-space. This transform allows
us to insert and also delete sites in a simple fashion by computing convex
hulls. Let M ⊂ S be any subset of k sites. M is transformed into a point
q(M ) in 3-space, by taking the centroid (center of mass) of M and lifting
110
Chapter 6. Higher Dimensions
it up vertically. More precisely,


1
q(M ) =
p,
pT p .
k
p∈M
p∈M
Now consider the set Qk (S) of all points that can be obtained from S in
this way. That is, Qk (S) = {q(M ) | M ⊂ S, |M | = k}.
Lemma 6.2. The part of the convex hull of Qk (S) that is visible from the
plane (i.e., its lower part) is dual to Vk (S).
The lemma can be proved by first mapping each subset M of k sites
into a non-vertical plane π(M ) in 3-space, given by
π(M ) : x3 =
2 T
1 T
x p−
p p
k
k
p∈M
p∈M
and then considering the upper envelope, Πk (S), of all these planes. It
is not difficult to show that the facets of Πk (S) project vertically to the
regions of Vk (S). The lemma then follows from observing the polarity (cf.
Section 6.2) between the planes π(M ) and the points q(M ).
Note the difference of embedding Vk (S) for k ≥ 2 in this way, compared
to its embedding into arrangement levels described earlier. The convex
polyhedral surface Πk (S) is not (the upper) part of the boundary of the
respective k-level — which rather is nonconvex — except for k = 1 where
both surfaces are the upper envelope of the same set of planes.
To construct Vk (S), we could just compute the point set Qk (S),
determine its convex hull, and then dualize its triangles, edges, and vertices
a point for each
that are visible from the plane. However, Qk (S) contains
n
k
subset of k sites of S, and thus has cardinality k = Θ(n ). On the other
hand, only O(k(n−k)) points lie on the convex hull, as Vk (S) has this many
regions.
We use randomized incremental insertion of sites in order to compute
this convex hull more efficiently. Let S = {p1 , . . . , pn }, and let Ci denote
the visible part of the convex hull of Qk ({p1 , . . . , pi }), for k + 1 ≤ i ≤ n.
Points of Qk (S) lying on the convex triangular surface Ci are called corners
of Ci .
We start by determining Ck+1 . Qk ({p1 , . . . , pk+1 }) contains k+1 points
which can be calculated in time O(k), so O(k log k) time suffices. The
generic step of the algorithm is the insertion of site pi into Ci−1 , for i ≥ k+2.
6.5. Higher-order Voronoi diagrams
111
(1) Identify all triangles of Ci−1 which are destroyed by pi and cut them
out. Let B be the set of corners on the boundary of the hole.
(2) Calculate the set P of all new corners created by pi .
(3) Compute the convex hull of P ∪ B, and fill the hole with the visible
part Γi of this convex hull. This gives Ci .
Each triangle ∆ of Ci−1 is dual to a vertex of Vk (S). This vertex is the
center of a circle that passes through three sites in S. Triangle ∆ will be
destroyed if this circle encloses pi . The destroyed triangles of Ci−1 form a
connected surface, say Σi−1 . Hence, if we know one such triangle in advance,
Σi−1 can be identified in time proportional to the number ni of its triangles.
Moreover, the set P of new corners can be calculated easily from the edges
of Σi−1 , as each such edge gives rise to a unique corner.
Lemma 6.3. Given Ci−1 we can construct Ci in time O(ni log ni ),
provided we know a triangle of Ci−1 that is destroyed by pi .
When looking for a starting triangle of Σi−1 , we profit from another
nice property of the duality transform: If the vertical projection ∆ of a
triangle ∆ of Ci−1 contains pi then ∆ is destroyed by the insertion of pi .
This leaves us with the problem of locating pi in the triangulation given by
the planar projection of Ci−1 .
In fact, we get the desired point-location structure nearly for free.
Adapting a technique used in Guibas et al. [390] for constructing Delaunay
triangulations, we do not remove the triangles of Ci−1 that get destroyed by
the insertion of pi , but mark them as old. When marked old, each triangle
gets a pointer to the newly constructed part Γi of Ci . The next site then is
located by scanning through the ‘construction history’ of Ci . The structure
for point location within each surface Γj , j ≤ i, which is needed in addition,
is a byproduct of the randomized incremental convex hull algorithm in
[390], which we used for computing Γj . (The latter structure is similar to
the Delaunay tree structure in Section 3.2, designed for the same purpose.)
In summary, the order-k Voronoi diagram for n sites in the plane can
be computed in expected time O(k 2 (n − k)) log n + nk log3 n), and optimal
(deterministic) O(k(n − k)) space, by an online randomized incremental
algorithm.
Full details and the following extensions are given in [102]. Deletion of
a site can be done in the reverse order of insertion, again by computing
a convex hull. The history-based point location structure used by the
algorithm can be adapted to support k-nearest neighbor queries (see
Subsection 8.1.1). A dynamic data structure is obtained, allowing for
insertions and deletions of sites in expected time O(k 2 log n + k log2 n),
112
Chapter 6. Higher Dimensions
and k-nearest neighbor queries in expected time O(k log2 n). This promises
a satisfactory performance for small values of k.
6.5.4. Cluster Voronoi diagrams
The order-k Voronoi diagram allots regions to subsets of fixed size k in the
underlying point set S. Let us mention, at the end of this section, a different
type of Voronoi diagram, whose regions are also defined by subsets of S. Let
t
C1 , . . . , Ct be a partition of S, that is, S = i=1 Ci and Ci ∩Cj = ∅ for i = j.
(Note that, by contrast, the subsets defining an order-k Voronoi diagram
do highly overlap.) We define the cluster Voronoi diagram for C1 , . . . , Ct
as the Voronoi diagram for these subsets (also called clusters) under the
Hausdorff distance
h(x, Ci ) = max{d(x, p) | p ∈ Ci }.
Cluster Voronoi diagrams have been mainly considered in the plane,
under different names, like the closest covered set diagram in Abellanas
et al. [5], or the min–max Voronoi diagram in Papadopoulou and Lee [581].
Sometimes also the name Hausdorff Voronoi diagram is used. Figure 6.10
gives an illustration.
By expressing distances to sites p ∈ S as planes π(p) as done earlier in
this section (see also Section 6.2), the diagram can be obtained by vertically
projecting the upper envelope of concave surfaces Σ(Ci ), i = 1, . . . , t, where
each surface Σ(Ci ) is the lower envelope of the planes in {π(p) | p ∈ Ci }.
Using this interpretation, Edelsbrunner et al. [304] bound the size of the
diagram by O(n2 α(n)), and give a construction algorithm running in the
same time bounds. (Here α(n) denotes the inverse of Ackermann’s function,
which is extremely slowly growing with n.) The size drops to O(n) if
the clusters are pairwise convex-hull disjoint. A lower bound is Ω(n2 ),
which is attained, for example, for 2-point clusters whose corresponding
line segments intersect heavily. The running time is O(n2 ) if only 2-point
or 1-point clusters are present.
Applying the same ‘Voronoi surface’ interpretation, Huttenlocher
et al. [415] show that the size of the cluster Voronoi diagram is only
O(nt · α(nt)), and that the diagram can be computed in time O(nt·log(nt)),
both size and runtime now being sensitive to the number t of clusters.
Alternatively, Papadopoulou [579] analyzes the size as O(n + γ), where
γ counts the number of piercing cluster pairs, of a certain type. She
also proposes a plane sweep construction algorithm, whose complexity
depends on several parameters. This algorithm was adapted later, in an
output-sensitive manner and also exploiting the 3D interpretation above,
6.5. Higher-order Voronoi diagrams
113
Figure 6.10. Voronoi diagram for 4 clusters under the Hausdorff distance. The region
of each cluster Ci is partitioned by the farthest-site Voronoi diagram of its defining sites
(i.e., by projected parts of the surfaces Σ(Ci )). The clusters are convex-hull disjoint in
this example. Note that sites interior to cluster hulls do not contribute to the diagram
and thus can be ignored.
in Xu et al. [709] to the case where clusters are (potentially intersecting)
isothetic rectangles, and the Hausdorff distance is taken in the L∞
metric. In fact, one obtains an additively weighted closest-site diagram (see
Section 7.4) for rectangles. This diagram is motivated by the problem of
critical area computation in VLSI design; see Papadopoulou and Lee [583].
Dehne et al. [250] gave an algorithm for the Voronoi diagram of
hull-disjoint clusters of total size n, which runs in O(n log4 n) time and
O(n log2 n) space. It is based on divide & conquer and therefore can be
parallelized efficiently. The main difficulty lies in dealing with the merge
chain in the conquer step, which can break into several, possibly cyclic,
components. This cannot happen for the classical Voronoi diagram, see
Section 3.3, which represents the special instance of singleton clusters
Ci = {pi } for i = 1, . . . , n. A recent randomized incremental approach,
in Cheilaris et al. [186], achieves an (expected) runtime more than a log n
factor faster, at optimal storage cost O(n).
Another concept related to order-k Voronoi diagrams are so-called
2-site Voronoi diagrams, considered in Barequet et al. [114]. The plane
is now divided into regions minimizing some measure of a point x with
114
Chapter 6. Higher Dimensions
respect to all possible pairs of sites {p, q} ⊂ S. This measure can be the
sum, product, or difference of the two distances d(x, p) and d(x, q), or the
area of the triangle xpq (or even others).
Interestingly, the first two choices just give the usual order-2 Voronoi
diagram, a structure of size O(n) which can be computed in O(n log n)
time.
Difference of distances, and triangle area, on the other hand, lead
to much more complex structures, of size O(n4+ε ). These upper bounds
stem, once more, from the complexity of envelopes of the respective (non4
linear) Voronoi surfaces. The latter bound can be sharpened to Θ(n
), by
n
exploiting a relation to so-called zones in an arrangement of 2 planes
in 3-space, one plane for each pair of sites. A zone, in this context, is the
collection of all cells in a (hyper)plane arrangement which are cut by placing
some additional hyperplane; see e.g. [300].
6.6. Medial axis in three dimensions
While the classical Voronoi diagram for point sites is well investigated
and understood in higher dimensions, knowledge becomes quite sparse for
sites of more general shape, concerning structural as well as algorithmic
properties. This is already true for the seemingly harmless generalization
from point sites to line segments or spheres.
The complexity of the Voronoi diagram for n line segments (and in
particular, the Voronoi diagram for straight lines) in Rd may be as large as
Ω(nd−1 ), as was observed by Aronov [67]. By a relation of Voronoi diagrams
to lower envelopes of hypersurfaces (see Subsection 7.5.1), the results in
Sharir [639] imply an upper bound of O(nd+ε ), for any ε > 0, on the size
of this diagram. Based on that relation, a numerically robust algorithm for
constructing the Voronoi diagram of straight lines in R3 has been designed
recently in Hemmer et al. [402].
Surprisingly, such types of Voronoi diagrams are still small ‘on the
average’. Dwyer [297] showed that the expected combinatorial complexity of
the Voronoi diagram of n uniformly distributed k-flats (affine k-dimensional
subspaces of Rd ) is O(nd/(d−k) ) for d ≥ 3. For example, the expected size
√
of the Voronoi diagram for n lines (1-flats) in R3 is only O(n n), and even
less in higher dimensions.
6.6.1. Approximate construction
No better bounds than Ω(n2 ) and O(n3+ε ) are known even in R3 , which
includes a case of particular interest in several applications, namely, the
6.6. Medial axis in three dimensions
115
medial axis of a (generally nonconvex) polyhedron P in three dimensions.
The arising bisectors contain, among other components, patches of
parabolic and hyperbolic surfaces and have a fairly complicated structure.
Regions are still simply connected (as in the plane, see Section 5.1), but even
for three straight lines as sites, the induced structure gets so intricate that a
separate paper has been devoted to its exploration; see Everett et al. [332].
The upper bound for the complexity of the diagram can be reduced to
O(n2+ε ) if the straight line sites are confined to have only constantly many
orientations; see Koltun and Sharir [470].
If the Euclidean distance function is replaced by some polyhedral
(convex) distance function (Section 7.2), then the upper bound can be
tightened to O(n2 α(n) log n), even if convex polyhedra (of size O(1))
are allowed as sites; see Chew et al. [217] and Koltun and Sharir [469],
respectively. A robust and practically efficient algorithm for computing the
medial axis of a boundary-triangulated object in 3-space under a convex
polyhedral distance function is described and implemented in Aichholzer
et al. [26]. Figures 6.11 and 6.12 give an illustration of the output, when a
tetrahedron is used as the unit ball for the convex distance function. Note
that the resulting medial axis is a piecewise linear object now, and that,
depending on the shape of the unit ball, the Euclidean medial axis will be
resembled to a certain extent.
We will elaborate on the approximate construction of the Euclidean
medial axis of a 3D object in this section — a notoriously difficult task
with various algorithms and techniques proposed over the years. As the
medial axis has evolved as a compact geometrical representation of shapes
Figure 6.11. A nonconvex polyhedron with triangulated (i.e. simplicial) boundary.
Figure 6.12. Piecewise linear medial axis under a tetrahedral distance
function of Figure 6.11.
116
Chapter 6. Higher Dimensions
in diverse applications, there is need for an efficient and stable computation
of this structure, or of suitable approximations thereof. Careful treatments
of medial axes are given, from the geometric and stability point of view, in
the survey article by Attali, Boissonnat, and Edelsbrunner [83], and from
the applied point of view, in the recent book by Siddiqi and Pizer [654].
The latter source also covers the case of the discrete medial axis, i.e., its
representation and computation as a binary (pixel or voxel) image; cf.
Section 11.1.
Computing the exact medial axis of a polyhedron in 3-space is
elaborate. A popular though time consuming approach are tracing methods,
see e.g. Milenkovic [542], Sherbrooke et al. [646], and Culver et al. [237]. The
edge skeleton of the medial axis is constructed piece by piece, basically at
the cost of looking at the whole polyhedron at each step. These continuous
methods do not require any sampling of the surface or volume of the input
object. However, they are prone to numerical errors, due to the inherent
algebraic complexity of the bisectors. Also, exactness turns out to be a
disadvantage in certain applications, as small and unwanted features of the
object (for example, small irregularities on the boundary) might get fully
reflected in its medial axis.
Sampling methods, on the other hand, do not suffer from these
drawbacks. In spatial sampling, the space of interest is subdivided into
suitable cells, and an approximation of the medial axis is computed at the
resolution of these cells; see Etzion and Rappoport [331] and Sud et al. [665].
This method is generally faster than continuous methods, though problems
arise with appropriately handling (almost) degenerate situations, unless
subdivision proceeds to a sufficiently fine level of detail.
Surface sampling approaches have received more attention. They try
to compute the medial axis via the Voronoi diagram (or the Delaunay
triangulation) of a set of sample points on the surface of the object. In
an early approach, Sheehy et al. [645] refine the Delaunay triangulation
of the sample points repeatedly, until the adjacency relationships defined
by the medial axis are determined. An interesting result in Amenta
and Kolluri [60], based on work in Amenta and Bern [58], shows that
if the ‘right’ Delaunay balls are chosen, and if the surface sample is
sufficiently dense, then the medial axis of the union of these balls is a
topologically correct approximation of the true medial axis of the object.
Related results, in Boissonnat and Cazals [140] and Amenta et al. [59],
concern the convergence of Voronoi vertices to the medial axis, and
the pruning of the Voronoi diagram with angle criteria (and others) in
order to obtain an approximation of the medial axis; see also Dey and
Zhao [271].
6.6. Medial axis in three dimensions
117
6.6.2. Union of balls and weighted α-shapes
Representing an object as a union of balls for approximate medial axis
construction is a flexible and natural method, as it resembles the geometric
definition of the medial axis as an infinite set of centers of maximal inscribed
balls; see Subsection 5.5.2. This infinite set, or the corresponding collection
of balls, is also called the medial axis transform (MAT) of an object, a
term common in computer graphics and object modeling; see e.g. Siddiqi
and Pizer [654].
The issues of how to choose an appropriate surface point sample
whose Delaunay balls serve as a candidate set, and how to keep small
the constructed set of balls, are treated among others in Amenta and
Kolluri [60] and in Aichholzer et al. [35], respectively. The resulting
algorithm simplifies the (topologically correct) medial axis automatically,
via the choice of the approximating balls, and delivers a piecewise linear
description. This method is also efficient, because the medial axis of a finite
union of balls can be computed comparatively easily, applying an elegant
structural result in Attali and Montanvert [86] and Amenta and Kolluri [61],
which we describe next.
Let B be a finite set of balls in R3 , and let U denote their union. By
interpreting the center of each ball b ∈ B as a point site p, and the squared
radius of b as a weight for p, we can consider the power diagram (Section 6.2)
of the resulting set of weighted sites.
We are interested in the faces of this power diagram that intersect
the union of balls U . Each such j-face f , for 0 ≤ j ≤ 3, corresponds to a
dual simplex σ(f ) in the respective regular tetrahedral complex, namely,
to the convex hull of the centers of the balls that define f . If general
position is assumed, then σ(f ) is a (3 − j)-simplex. Tetrahedra, triangles,
line segments, and single points may occur as dual simplices. The collection
of these simplices is called the weighted α-shape, A(B), of the set B of
balls, a concept introduced in Edelsbrunner [302] as a means for molecular
modeling and shape reconstruction. See Figure 6.13 for a picture in one
dimension lower.
A(B) consists of various components, the full-dimensional ones (of
dimension three in our case), and components of dimensions two, one, and
zero. A face of A(B) is termed singular if it is not part of any tetrahedron
that contributes in forming the α-shape. Note that singular faces thus
cannot be full-dimensional.
Theorem 6.8. The medial axis of U =
b ∈ B b is composed of the
singular faces of A(B), plus the portion of the Voronoi diagram of U ’s
vertices that resides inside the non-singular parts of A(B).
118
Chapter 6. Higher Dimensions
Figure 6.13. Power diagram for nine disks
and their weighted α-shape. In this twodimensional example, three line segments
and three points (1 being isolated) occur as
singular faces.
Figure 6.14. The medial axis of the
union of these disks. Singular faces
are copied, and non-singular parts are
partitioned by the Voronoi diagram of
the union’s vertices.
Figure 6.14 gives an illustration. Theorem 6.8 does not only lead to
construction methods for the medial axis of the union of balls in general
dimensions (via power diagrams and Voronoi diagrams; Section 6.2), but it
also makes explicit that this medial axis has a piecewise linear structure.
Surprisingly though, its combinatorial complexity is unsettled; a trivial
upper bound is O(n4 ), stemming from the O(n2 ) vertices of the balls’ union.
We refer the interested reader to the survey paper on the union of geometric
objects by Agarwal et al. [15].
The level of approximation to the real medial axis (of the original 3D
object) can be tuned by the number of Delaunay balls used; see Aichholzer
et al. [35] for details.
In such a medial ball representation of an object, the Delaunay balls
will usually intersect heavily. In the 2D case, Giesen et al. [364] could
take advantage of this fact. They proved that, if the boundary of a planar
shape P is sampled sufficiently densely, then the vertices of the resulting
union U of Delaunay disks all need to be sample points. Consequently,
the medial axis of U just consists of the edges and vertices of the sample
points’ Voronoi diagram that lie entirely inside P . It suffices to compute
one Voronoi diagram, rather than the regular triangulation of the disks and
the parts of the Voronoi diagram within its (non-singular) triangles.
In Subsection 8.2.2, an unweighted variant of α-shapes will be
considered. In fact, if all ball radii are equal to a fixed value α, their power
diagram is just the Voronoi diagram, V (S), of the set S of their centers.
6.6. Medial axis in three dimensions
119
Now, if a face f of V (S) intersects the boundary ∂ U of their union U ,
any point x ∈ f ∩ ∂ U is the center of a sphere of radius α that passes
through f but does not enclose any other element of S. That is, the dual
face σ(f ) (which is a face of A(B)) belongs to the Delaunay tetrahedrization
DT(S), and σ(f ) is also part of the (unweighted) α-shape of the point set S
according to its definition in Subsection 8.2.2.
This implies that the α-shape of a point set constitutes the boundary
faces of the weighted α-shape for congruent radius-α balls. In R2 , for
example, it consists of all singular edges, plus all edges bounding the nonsingular components.
Recall from Subsection 6.2.1 that the union of n balls is decomposed
nicely by their power diagram. In R2 , this leads to an O(n log n) algorithm
for computing the union of n disks [107, 90] — even though their
bounding circles may intersect in Θ(n2 ) points. The area of their union
can be calculated in additional O(n) time. By contrast, the fastest known
algorithm for computing the area of the union of n triangles runs in
quadratic worst-case time. The latter task is one of the so-called n2 -hard
problems listed in Gajentaan and Overmars [347]. Another important
n2 -hard problem is testing a given set of n points in the plane for general
position, with respect to collinearities. These problems are unlikely to have
a subquadratic solution, and do so only if the following basic arithmetic
problem (the 3SUM problem) does: Given a set M of n integers, do there
exist numbers a, b, c ∈ M with a + b + c = 0?
An intuitive reason why disks are easier to treat is that they are
homothets of each other, while triangles in general are not. Another
explanation is that disks and balls are fat objects, i.e., their ratio of diameter
and width (size of smallest enclosing slab) is bounded by a constant (1 in
that case). For fat convex objects, more efficient algorithms can be designed
in many cases; see e.g. de Berg et al. [243] and Agarwal et al. [15].
Let us mention at this place that another flexible and versatile tool for
making explicit the interaction between the balls in a finite set B can be
based on power diagrams.
Generalizing to R3 a concept introduced in Cheng et al. [203],
Edelsbrunner [303] defines the -skin surface for B as the envelope of an
infinite set of balls derived from B by convex combination and shrinking
(by a factor of ). The family of all -skin surfaces, for ∈ [0, 1], can be
represented by a cell complex C in a horizontal strip in R4 , defined by
‘linearly interpolating’ between the regular simplicial complex of B situated
in the hyperplane x4 = 0, and the power diagram of B situated in the
hyperplane x4 = 1. Now, intersecting C with x4 = gives a complex of
convex three-dimensional cells — either shrunk copies of regular simplices
120
Chapter 6. Higher Dimensions
or power cells, or polyhedra filling the gaps between. These cells decompose
the -skin surface into spherical and hyperbolic patches.
Skin surfaces are smooth manifolds that vary continuously with the
centers and radii of the balls. Application in molecular modeling and shape
morphing are described in [303]. In particular, they allow to define a
‘morphing distance’ between shapes which, in the 2D case [203], can be
used for automatic pattern recognition (e.g., for the classification of Chinese
characters).
Constructing the family of power diagrams that arise during the
morphing process is discussed in Chen and Cheng [199].
6.6.3. Voronoi diagram for spheres
Let us observe that the medial axis of a finite union U of balls is not a subset
of the seemingly similar Voronoi diagram for their n bounding spheres. The
former structure is defined by distances to the boundary ∂U of U , whereas
the latter structure is based on distances to the individual spheres, and is
of interest and importance it its own right.
The Voronoi diagram for spheres is a tessellation of the entire space,
and is non-linear also in the exterior of its defining spherical sites. The
bisector of two spheres is a sheet of a hyperboloid of rotation, bending
towards the smaller sphere. (If both spheres have the same radius, their
bisector is a hyperplane.) The diagram is composed of hyperbolic and
possibly linear patches and has quite a complicated combinatorial structure.
Though regions are star-shaped — as each of them is visible from the
respective sphere center — they typically are multiply adjacent, and
boundary facets may arise that are not simply connected.
In many applications, the focus is on the part of the diagram exterior
to U . Given its importance in practice, for example in molecular biology
(Will [706] and Angelov et al. [63]) and in material sciences (Goede
et al. [365] and Naberukhin et al. [560]), the Voronoi diagram for n spheres
in Rd has received only moderate attention.
A size of O(nd/2+1 ) can be shown, by its relationship to power
diagrams in Rd+1 proved in Aurenhammer and Imai [98], which can also
be used to construct the diagram efficiently. More recently, algorithms
similar in spirit but more convenient, have been devised in Will [706],
Boissonnat and Karavelas [146], and Boissonnat and Delage [141]. They
construct the diagram region by region, exploiting that a single region
corresponds to a certain convex hull of spheres, which in turn can be
computed by intersecting a power diagram with the unit sphere in one
dimension higher. Interestingly, and unlike the classical Voronoi diagram or
6.6. Medial axis in three dimensions
121
the power diagram, a single region can be as complex as the entire diagram,
hence of size Θ(nd/2+1 ) in d ≥ 3 dimensions, giving Θ(n2 ) in R3 .
Different methods of construction exist for d = 3, being more friendly
to implementation though theoretically slower; see Kim and Kim [453] and
references therein. They first construct the classical Voronoi diagram for
the centers of the spheres, and then let the centers gradually expand one
by one into spheres of the required sizes, thereby updating their regions in
the diagram.
Note that the Voronoi diagram for spheres is just the additively weighted
(Euclidean closest-site) Voronoi diagram with the sphere centers as its sites,
the weights being the sphere radii. Its two-dimensional variant, the Voronoi
diagram for n circles, is of linear size because it consists of n star-shaped
planar regions. We discuss this useful structure, along with several of its
applications, in Section 7.4.
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Chapter 7
GENERAL SPACES & DISTANCES
So far we have mainly discussed Voronoi diagrams of sites in d-space that
are defined with respect to the Euclidean distance function. Now we want
to generalize the space in which the sites are situated and/or the distance
measure used; but we shall only discuss the case of point sites, apart from
a few important exceptions.
The main questions are which of the structural properties the standard
Voronoi diagram enjoys will be preserved, and if the remaining properties
are strong enough to apply one of the algorithmic approaches introduced
in Chapter 3 for computing the Voronoi diagram.
7.1. Generalized spaces
7.1.1. Voronoi diagrams on surfaces
Since the surface of earth is not flat, it seems very natural to ask about
Voronoi diagrams of point sites on curved surfaces in 3-space. The distance
between two points on the surface is the minimum Euclidean length of a
curve that connects the points and runs entirely on the surface. Such a
curve will be called a shortest path or a geodesic path.
Brown [170] has addressed the Voronoi diagram for point sites on the
surface of a sphere in 3-space. Here great circles play the role of lines in
the Euclidean plane. In fact, the bisector of two points is a great circle,
and the shortest paths are segments of great circles, too. (One can show
that the only other metric space in which all bisector segments are shortest
paths is the hyperbolic space; see Busemann [174].)
For each pair of antipodal points on the sphere there is a continuum
of shortest paths connecting them. But this does not affect this (geodesic)
Voronoi diagram of n points; it can be computed in optimal O(n log n) time
123
124
Chapter 7. General Spaces & Distances
and linear space, by adaption of the algorithms mentioned in Chapter 3;
see, in particular, the paper by Na et al. [559].
Given a line and a point p not on , in Euclidean geometry one can
draw exactly one line through p that does not intersect , the line parallel
to . In hyperbolic geometry many such lines exist. One model of hyperbolic
geometry named after Felix Klein is given by the interior of the unit sphere;
lines correspond to chords, and the distance between two points p and q is
given by
1−p·q
.
d(p, q) = arccosh (1 − |p|2 )(1 − |q|2 )
Nielsen and Nock [563] have observed that in this model bisectors are
hyperplanes. Consequently, by the results in Subsection 6.2.3, hyperbolic
Voronoi diagrams in Klein’s model are power diagrams. Nilforoushan and
Mohades [564] and, very recently, Bogdanov et al. [138] have considered the
Poincaré model of hyperbolic geometry, where lines correspond to diameters
and to circular arcs meeting the boundary of the unit sphere at a right angle.
The latter paper discusses how to extract from the Euclidean Delaunay
complex those simplices forming the hyperbolic Delaunay complex, in
general dimensions d. Remarkably, these simplices can be computed with
rational arithmetic for rational input points, a property well suited for
implementation. That the hyperbolic Voronoi diagram can be derived from
its Euclidean counterpart has first been observed in Devillers et al. [265].
It can be constructed in O(n log n) time and O(n) space in the twodimensional case.
Applications arise, for example, in meshing of high-genus surfaces, in
information theory, and in routing on wireless ad hoc networks. See Tanuma
et al. [679], who also consider several generalizations, e.g., for spheres and
segments as sites.
Quite different is the situation on the surface of a cone, which we will
discuss next. In order to determine the bisector of two points p and q, we
can cut the cone along a halfline emanating from the apex, and unfold
it; in Figure 7.1 the halfline diametrically opposed to p has been chosen.
Since curve length does not change in this process, each shortest path on
the cone that does not cross the cut is transformed into a shortest path
in the plane, i.e., into a line segment. In order to represent those shortest
paths that cross the cut, we add to the unfolded cone two more copies,
as shown in Figure 7.1(ii). Now the shortest path on the cone from some
point x to site q corresponds to the shortest one of the line segments qx,
q x, and q x. This explains why the unfolded bisector B(p, q) consists of
segments of the planar bisectors of p, q and p, q .
7.1. Generalized spaces
125
p
q’
q
q’’
p’
p’’
q’
(i)
Figure 7.1.
p
p
q
p
q
(ii)
A cone sliced and unfolded, showing the bisector of p and q.
In spite of this strong connection to the plane, the Voronoi diagram of
points on a cone has structural properties surprisingly different from the
planar Voronoi diagram. If the unfolded cone forms a wedge of angle less
than 180◦ then the bisector of two points can be a closed curve. If three
points p, q, r are placed, in this order, on a halfline emanating from the
apex of such a cone, the bisector B(q, r) fully encircles B(p, q) which in
turn encircles the apex. This causes the Voronoi region of q in V ({p, q, r})
to be not simply connected.
Also, two Voronoi regions can be adjacent in more than one Voronoi
edge. Such a situation is shown in Figure 7.2, on the unfolded cone. The
bisectors of the three points cross twice, at the Voronoi vertices v and w;
the latter happens to lie on the cut. (It is interesting to observe that none
of these phenomena occurs on the sphere, although there, too, bisectors are
closed curves and cross twice.)
Despite its more complex structure, the Voronoi diagram of n points
on the surface of a cone can be constructed in optimal time O(n log n) and
space O(n), using a sweep circle that expands from the apex; see Dehne
and Klein [247]. This approach works without unfolding the cone.
Mazón and Recio [526] have independently pointed out the algebraic
background of the unfolding procedure illustrated by Figure 7.1(i), and
obtained the following generalization. Let P denote the Euclidean plane
or the sphere, and let G be a discrete group of motions on P , i.e., a
group of bijections that leave the distance between any pair of points of
Chapter 7. General Spaces & Distances
126
p
p
q
p
r
w
v
r
q
rq
r q
Figure 7.2.
The common border of the Voronoi regions of q and r consists of two edges.
P invariant. We assume for G that, for each point p ∈ P , there exists a
constant c satisfying
p = g(p) =⇒ d(p, g(p)) ≥ c
for all motions g ∈ G.
Examples in the plane are the group generated by a rotation of rational
angle about some given point, or the group generated by two translations
that move each point a fixed distance to the right and a fixed distance
upwards, respectively. In the 19th century, mathematicians have completely
classified all discrete groups.
Two points p, p ∈ P are equivalent if there exists a motion in G
that takes p to p ; the equivalence class, [p], of p is called the orbit of p.
The quotient space, P/G, consists of all orbits. In order to geometrically
represent P/G one starts with a connected subset of P that contains
a representative out of every orbit; equivalent points must be on the
boundary. Such a set is called a fundamental domain, if it is convex. The
following lemma provides a nice way of obtaining a fundamental domain;
a proof can be found in Ehrlich and Im Hof [318].
Lemma 7.1. Let p be a point of P that is left fixed only by the unit element
of G. Then its Voronoi region VR(p, [p]) is a fundamental domain.
In Figure 7.1(ii), for example, the point set {p, p , p } is the orbit of p
under a clockwise rotation by 120◦ . The Voronoi region VR(p, {p, p , p })
equals the master copy of the unfolded cone, as drawn by solid lines. Each
interior point x is the only point of [x] contained in this region, only the
7.1. Generalized spaces
127
points on the boundary (i.e., the cut) of the unfolded cone are mapped into
each other by rotation. If we identify these two halflines we obtain the cone
depicted in Figure 7.1(i), a model of the quotient space of the Euclidean
plane over the cyclic group of order 3.
In a similar way we would obtain a rectangle as the fundamental domain
of the group of two translations mentioned above, and identifying opposite
edges would result in a torus.
If we want to compute the Voronoi diagram of a set S of point sites
on a surface associated with such a quotient space P/G, we can proceed as
follows. Let S0 denote a set of representatives of S in a fundamental domain
D ⊂ P . First, we compute the Voronoi diagram V ([S0 ]), where [S0 ] denotes
the union of the orbits of the elements of S0 , an infinite but periodic set.
Due to [526], V ([S0 ]) can be obtained by applying the motions in G to the
Voronoi diagram of a finite set of points of S0 and translated copies of S0 .
Theorem 7.1. There exists a finite subset U of [S0 ] such that the identity
V ([S0 ]) = [V (U ) ∩ D] holds.
If one removes from V ([S0 ]) all Voronoi edges that separate points of
the same orbit and intersects the resulting structure with the fundamental
domain D, the desired diagram V (S) results, after identifying equivalent
points. Although the set S0 can be constructed effectively, it seems hard to
establish an upper bound for the efficiency of this step.
Ehrlich and Im Hof [318] have studied, from a differential geometrist’s
point of view, structural properties of the Voronoi diagram in such
Riemannian manifolds where any two points are connected by a unique
shortest path.
Of particular interest, though of different flavor, is the discrete case of a
polyhedral (i.e., polygonal) surface in 3-space, of combinatorial size m. Here,
constructing Voronoi diagrams is closely related to computing single-source
shortest paths by the continuous Dijkstra technique. (For Dijkstra’s method,
and graph search methods in general, see e.g. the book by Gibbons [363].)
If we place those points of a surface patch in the same region whose shortest
paths to the source have the same combinatorial structure (i.e., they
visit the same edges and vertices in the same order) then Voronoi-like
decompositions result. Conversely, the continuous Dijkstra algorithm can
often be implemented to run efficiently for n sources simultaneously, so
that for each point on the surface the nearest source is known; see, e.g.,
Schreiber and Sharir [622] for a recent result.
In general, the continuous Dijkstra technique allows the Voronoi
diagram of n sites to be computed in O(N 2 log N ) time and O(N 2 )
128
Chapter 7. General Spaces & Distances
space, where N = max{m, n}; see Mitchell et al. [547]. Moet et al. [549]
and Aronov et al. [68] have studied the complexity of bisectors and
Voronoi diagrams on triangulated terrains. It turns out that, under realistic
assumptions on the size and slope of the m terrain triangles, and on the
density of their projections, bisectors consist of only O(m) segments, and
√
Voronoi diagrams are of complexity O(n + m n). Driemel et al. [287]
recently even improved this to linear, O(n + m), if uniform distribution
of the n sites is assumed.
An algorithm for computing the Delaunay triangulation on the surface
of a compact polyhedron in 3-space is developed in Indermitte et al. [423],
along with a possible modeling application in biology. They propose a
flipping procedure and show that it halts after a finite number of flip
operations.
7.1.2. Specially placed sites
The above setting (geodesic Voronoi diagrams on surfaces) should not be
confused with the scenario where only the sites are confined to lie on a
specific surface, and the classical Euclidean Voronoi diagram (or Delaunay
triangulation) in 3-space is sought. The latter scenario arises in the
study of realistic input models, and is motivated by surface reconstruction
algorithms that compute the Delaunay triangulation of a set of sample
points.
For polyhedral surfaces, the complexity of the 3D Delaunay triangulation is O(n polylog n) for random samples, when sites are uniformly
distributed on the surface, see Golin and Na [368], in contrast to the
O(n) complexity for uniform distribution in 3-space, see Dwyer [296]. (The
function polylog n stands for (log n)c , for some constant c > 1.) For random
samples on certain surfaces of bounded curvature, including cylinders of
√
constant radius and height, a complexity of O(n n log n) is shown in
Erickson [328]. The case of cylinders is settled in Devillers and Goaoc [263],
with a tight bound of Θ(n log n).
As a recent result in Rd , Amenta et al. [57] prove that for n point sites
‘well distributed’ on the boundary of a k-dimensional nonconvex polyhedron
(2 ≤ k ≤ d − 1), the size of the Delaunay simplicial complex is O(n(d−1)/k ).
They even improve this bound slightly, to asymptotic tightness.
A flipping algorithm for the 2D Delaunay triangulation on the flat
torus (i.e., periodic copies of the square, cf. Subsection 7.1.1) is described
in Telley [681]. The 3D Delaunay triangulation on the flat torus (periodic
cube) is treated algorithmically in Caroli and Teillaud [183]. They adapt the
incremental site insertion method, and mention various application areas,
7.2. Convex distance functions
129
including astronomy, material engineering, biomedical computing, and fluid
dynamics.
In fact, Voronoi diagrams for periodically placed sites have been among
the very first instances considered. Carl Friedrich Gauss, Gustav Lejeune
Dirichlet, and later Georgi Feodosjewitsch Voronoi [693, 694] observed that
quadratic forms have an interpretation in terms of Voronoi diagrams for
parallelohedrally ordered sites in space. Moreover, certain tilings of space by
convex polyhedra, generated by motions that form a crystallographic group,
can be interpreted as Voronoi diagrams. Their regions are all congruent
copies of each other, and have been called special fundamental domains by
Arthur Schönflies [620] and plesiohedra by Boris Delone (Delaunay) [252].
In other words, such congruent ‘space-filler ’ polytopes can be found by
means of computing Voronoi diagrams for carefully chosen sites. There has
been a hunt for space-fillers with many facets over the years. We refer to
the survey articles on tilings and geometric crystallography by Schulte [623]
and Engel [320] for more material on this topic.
An important special case are Voronoi diagrams whose sites form a
lattice. A lattice in d-space Rd is specified by its basis, i.e., a set of vectors
{v1 , . . . , vd } that span Rd . The sites are obtained by integer combinations
of the vectors from the lattice basis. The regions of such a Voronoi diagram
are all translates of each other; they are centrally symmetric (convex)
polytopes in Rd . Thus, it is meaningful to talk of the Voronoi region of
a given lattice — also called its prototile — or of its dual lattice Delaunay
polytope.
Constructing the prototile VR(L) of a lattice L basically amounts to
singling out its neighboring sites. This is a complicated task in higher
dimensions; computing the number of vertices of VR(L) is an #P-hard
problem. See Dutour et al. [294], who also give an algorithm that works
efficiently in low dimensions. Earlier construction methods can be found in
Viterbo and Biglieri [692]. Given its prototile, many parameters of a lattice
can be determined. For example, its packing radius equals the inradius
of VR(L), and its covering radius equals the circumradius of VR(L). Main
applications of Voronoi diagrams for lattices arise in information theory
and cryptography. Lattice Delaunay polytopes for positive quadratic-form
distances (Subsection 7.4.5) are treated in Dutour and Rybnikov [293]; see
also [692, 294] for further references and applications.
7.2. Convex distance functions
In numerous applications the Euclidean metric does not provide an
appropriate way of measuring distance. In the following sections, we
Chapter 7. General Spaces & Distances
130
consider the Voronoi diagram of point sites under distance measures
different from the Euclidean. We start with convex distance functions,
a concept that generalizes the Euclidean distance but slightly. Whereas
this generalization does not cause serious difficulties in the plane, surprising
changes will occur as we move to 3-space.
7.2.1. Convex distance Voronoi diagrams
Let C denote a compact convex set in the plane that contains the origin in
its interior. Then a convex distance function can be defined in the following
way. In order to measure the distance from a point p to some point q, the
set C is translated by the vector p. The halfline from p through q intersects
the boundary of C at a unique point q ; see Figure 7.3. Now one puts
dC (p, q) =
d(p, q)
.
d(p, q )
By definition, C equals the unit disk of dC , that is, the set of all points
q satisfying dC (0, q) ≤ 1. Sometimes, C is also referred to as the shape
that defines dC (cf. Subsection 7.2.2), or as the calibration figure or
Eichfigur (a name used in Hermann Minkowski’s work of 1927 on number
theory [544]). The value of dC (p, q) does not change if both p and q are
translated by the same vector.
One can show that the triangle inequality dC (p, r) ≤ dC (p, q) + dC (q, r)
is fulfilled, with equality holding for collinear points p, q, r. By definition,
we have dC (p, q) = dC (q, p), where C denotes the reflected image of C
about the origin. We can define the Voronoi diagram based on an arbitrary
convex distance function by associating, with each site p, all points x of the
plane such that dC (p, x) ≤ dC (q, x) holds for all other sites q.
If the set C is (point-) symmetric about the origin then dC is called a
symmetric convex distance function. This is equivalent to saying that the
function q → dC (0, q) is a norm in the plane.
q
o
Figure 7.3.
p
q’
Defining a convex distance function dC with unit disk C.
7.2. Convex distance functions
131
Well-known is the family of Lp (or Minkowski) norms, 1 ≤ p < ∞,
defined by the equation
Lp (q, r) =
p
p
p
|q1 − r1 | + |q2 − r2 | ,
for q = (q1 , q2 ), r = (r1 , r2 ).
Whereas L2 is the Euclidean distance, the L1 -norm L1 (q, r) = |q1 − r1 | +
|q2 − r2 | is called the Manhattan distance of q and r, because it equals the
minimum length of a path from q to r that follows a rectangular grid. Its
unit circle is shown in Figure 7.4(i). If index p tends to ∞, the value of
Lp (q, r) converges to the L-infinity norm, L∞ (q, r) = max{|q1 − r1 |, |q2 −
r2 |}, also called the maximum norm. The unit circle of L∞ equals the
aligned square [−1, 1]2; therefore, L1 and L∞ are related by a 45◦ rotation
of the plane.
In Figure 7.4(ii) the bisector of two points under the Manhattan
distance L1 is shown. Another possible situation can be obtained by rotating
picture (ii) by 90◦ . But if p and q are the diagonal vertices of an aligned
square then their bisector B(p, q) is no longer a curve: it consists of two
quarterplanes connected by a line segment; see Figure 7.4(iii).
If the unit disk C is strictly convex, that is, if its boundary contains
no line segment, this phenomenon cannot occur. In fact, we can construct
the bisector of p and q in the way depicted in Figure 7.5. Let T and B
denote the outer tangents to the two copies Cp , Cq of C, and let t, t and b, b
be the respective tangent points. Consider a parallel line L between T and
B. It intersects the boundaries of Cp , Cq in four points. Two of them, l and
l , are facing each other. The halflines from p through l and from q through
l intersect in a point of the bisector B(p, q), and each point of B(p, q)
can be obtained this way. Thus, line L moving from T to B establishes a
1
q
0
1
p
q
p
B(p,q)
B(p,q)
(i)
Figure 7.4.
(ii)
(iii)
Unit circle and bisectors of the Manhattan distance L1 .
132
Chapter 7. General Spaces & Distances
Figure 7.5.
Constructing the bisector of p and q.
homeomorphism of B(p, q) and the boundary part of Cp between t and b,
which is homeomorphic to a line. Therefore, B(p, q) is homeomorphic to a
line, too.
We observe that the above argument would fail if C were flat at the
top, so that t and t were contained in horizontal line segments. Indeed, as
Cp and Cq expand, these segments would, at some time, overlap, and their
intersection would write out a two-dimensional bisector part. This happens
in the situation shown in Figure 7.4(iii). To avoid this phenomenon, one
can (symbolically) perturb the given point sites in such a way that no two
of them define a line parallel to a line segment in the boundary of C. In
Figure 7.4(iii), for example, if we imagine p and q to be slightly moved
apart in x-direction, the polygonal curve drawn by three bold lines results
as their bisector.
Alternatively, one can assume that the point sites in S are linearly
ordered. If p ≺ q then the common boundary of the dominance
domain D(q, p) and B(p, q) is chosen as a bisecting curve for p and q,
i.e., the set B(p, q) is added to the region of p with respect to q. This choice
7.2. Convex distance functions
133
leads to a consistent definition of Voronoi diagrams. We shall employ it in
Definition 7.5 in Section 7.3.
Although the bisector of two points can be described quite easily, it may
have a complicated shape. Corbalan et al. [233] gave an example of a strictly
convex set C, with a smooth boundary, and of two points p, q whose bisector
under dC ‘wiggles’ infinitely often between two parallel lines. (An object is
called smooth if it has a unique tangent everywhere.) By slightly translating
p and q, one obtains two bisector curves, B(p, q) and B(p , q ), that cross
infinitely often. When constructing Voronoi diagrams, some care is needed
in dealing with this complication.
But even though the bisector B(p, q) may wiggle, it is still confined to
the bent strip defined by the halflines from p through t and b, and from q
through t and b ; see Figure 7.5 for notation.
For each strictly convex distance function, the strict triangle inequality
holds: We have dC (p, r) < dC (p, q) + dC (q, r) unless p, q, r are collinear.
Moreover, two circles with respect to dC intersect in at most two
points (thus being so-called pseudo-circles), and two related bisectors
B(p, q), B(p, r) intersect in at most one point; proofs can be found in Icking
et al. [418].
Voronoi regions based on convex distance functions are in general not
convex, as the example of the Manhattan distance shows. But they are still
star-shaped, as seen from their sites: For each point x ∈ D(p, q), the line
segment px is also contained in D(p, q). This follows from a more general
fact shown in Lemma 7.4 later.
The star-shapedness of the Voronoi regions, together with the convexity
of the circles, is a property strong enough for applying the divide & conquer
algorithm; cf. Section 3.3. Hwang [416], Lee [493], and Lee and Wong [499]
have studied the Voronoi diagram based on Lp norms (sometimes called
Lp -Voronoi diagram for short); they provided algorithms that run within
O(n log n) many steps. Here and in the sequel, a step not only denotes a
single operation of a Real Random Access Machine (cf. [596]) but also an
elementary operation like computing the intersection of two bisector curves.
Widmayer et al. [704] have described an optimal algorithm for
computing the Voronoi diagram of a distance function based on a convex
m-gon, C. This generalizes the Manhattan distance to an environment
where motions are restricted to a finite set of orientations, given by the
rays from the origin through the vertices of C. The diagram has a size of
O(mn) in this case, if individual line segments on bisectors are counted as
edges.
Eventually, Chew and Drysdale [215] studied two-dimensional Voronoi
diagrams based on general convex distance functions dC . Aiming for an
Chapter 7. General Spaces & Distances
134
O(n log n) divide & conquer algorithm, one crucial point is in the merge
step. If we use a split line to subdivide the site set S into subsets L and R,
the merge chain B(L, R), consisting of all edges of the Voronoi diagram
VdC (S) that separate L-regions from R-regions, need not be connected.
Fortunately, each of its connected components turns out to be an unbounded
chain, not a loop. Therefore, a starting edge of each component can be found
at infinity, as in the Euclidean case; see Section 3.3.
It is well known that each symmetric convex distance function dC
is equivalent to the Euclidean distance d, in the sense that for suitable
constants a and A, the inequalities
a · d(p, q) ≤ dC (p, q) ≤ A · d(p, q)
hold for all points p, q. In particular, a sequence of points pi converges to
some limit point p under dC iff it does so under the Euclidean distance.
One might wonder if these similarities cause the Voronoi diagrams of
d and dC to have similar combinatorial structures. A counterexample is
shown in Figure 7.6; the regions of p and q are adjacent under L2 but not
under L1 .
Corbalán et al. [232] have provided systematic answers to the above
question. Let dC , dD denote two symmetric and strictly convex distance
functions whose unit circles are smooth. If for each set S of at most
four points in the plane, VdC (S) and VdD (S) have the same structure as
embedded planar graphs, then D must be a scaled version of C. More
generally, if for each set S of at most four points the Voronoi diagram
p
q
p
q
Figure 7.6. Four point sites whose Voronoi diagrams based on the Manhattan and the
Euclidean distance have a different combinatorial structure.
7.2. Convex distance functions
135
VdC (S) has the same combinatorial structure as VdD (f (S)), for some
bijection f of the plane, then f is linear and f (C) = D holds, up to scaling.
Conversely, if f is a linear bijection of the plane, then the Voronoi diagram
of f (S) with respect to the convex set f (C) can be obtained by applying f
to VdC (S), for each site set S.
The image of the Euclidean circle under a linear bijection is an ellipse.
The above result implies that convex distance functions based on ellipses are
the only ones whose Voronoi diagrams can be obtained from the Euclidean
Voronoi diagram of a transformed set of sites. The following theorem gives
an even stronger reason why such a reduction is not possible for unit circles
other than ellipses.
Theorem 7.2. Let C be a strictly convex, compact set, symmetric about
the origin, which is not an ellipse. Then there exists a set of nine points
in the plane whose Voronoi diagram with respect to dC has a structure no
Euclidean Voronoi diagram can achieve.
In the proof given in [232], a Voronoi diagram based on dC is
constructed whose dual tessellation (see Subsection 7.2.2) is either of the
topological shape shown in Figure 7.7(i), or of similar type. Let us assume
that it can be realized by a Euclidean Delaunay tessellation; cf. Chapter 2.
By construction, r lies outside the circumcircle of q, s, z, so that we have
β + β < π; see Figure 7.7(ii). Similarly, δ + δ < π holds. Since p lies on
the circumcircle of w, q, z we have α + α = π, and γ + γ = π holds for the
same reason. The primed angles at z add up to 2π. Therefore, we obtain
α + β + γ + δ < 2π.
q
p
α
r
s
β
β’ α’
γ’
q
r
β
w
δ’
s
β’
Z
Z
γ
δ
t
v
u
(i)
(ii)
Figure 7.7. The graph in (i) is not a Euclidean Delaunay tessellation. In (ii) we have
β + β < π iff r lies outside the circle. Equality holds iff r lies on the circle.
136
Chapter 7. General Spaces & Distances
But this is impossible because the points different from z must be in convex
position (because the convex hull of a point set equals the boundary of the
unbounded face of its Delaunay triangulation), so that each of the angles at
p, r, t, v includes an angle of the quadrilateral drawn by dashed lines. These
angles add up to 2π.
7.2.2. Shape Delaunay tessellations
The Delaunay tessellation of a set S of sites can be defined in two ways:
As the geometric dual of the Voronoi diagram of S, if we connect two sites
in S by a dual edge iff their Voronoi regions are adjacent; or by using the
empty circle property: An edge pq between two sites p, q ∈ S is a Delaunay
edge iff there exists some circle that passes through p and q but encloses
no other site in S.
For the Euclidean distance, these two definitions are equivalent.
However, for general convex distance functions dC , this is no longer true,
because the symmetry property is not guaranteed. More specifically, in the
definition of the bisectors B(p, q) of the diagram VdC (S) (Subsection 7.2.1),
distances from the sites are measured, whereas for the empty-circle
property, distances are taken from the center of the convex shape C,
i.e., towards the sites. Since we have dC (p, q) = dC (q, p), where C denotes the reflected image of C about its center, the dual of VdC (S) is
just the (edge-based) ‘empty-circle’ Delaunay tessellation for the shape C .
This fact has a surprising consequence: The dual of VdC (S), and thus,
the combinatorial structure of VdC (S), is invariant under movements
of the center of C. The geometric structure changes, of course.
Drysdale [288] considered such so-called shape Delaunay tessellations,
and showed how the divide & conquer approach in Guibas and Stolfi [394]
(see also Section 3.3) can be extended to run, in optimal time
O(n log n), for general convex distance functions. Ma [514] proposed an
incremental algorithm that applies flips to repair the tessellation after
the insertion of each site (cf. Section 3.2). Skyum [656] constructs the
shape Delaunay tessellation, based on a smooth and strictly convex distance
function, using the sweep line approach (Section 3.4), in O(n log n) many
steps.
Shape Delaunay tessellations have several interesting applications.
For example, they lead to high-quality and dynamic spanner graphs
(Subsection 8.2.4). They undergo only a nearly quadratic number of discrete
changes when the sites are moved on algebraic trajectories of constant
description; see Agarwal et al. [12], and Section 9.1 on Voronoi diagrams
for moving sites.
7.2. Convex distance functions
137
e
Figure 7.8. A shape Delaunay tessellation where C is an obtuse triangle (whose copies
are shown lightly shaded). The support hull is emphasized. Note the direction-sensitivity
of the tessellation’s triangles according to C.
If the shape C is not strictly convex and smooth, then the shape
Delaunay tessellation, for short DTC (S), is not a full triangulation of the
convex hull of S. Consult Figure 7.8: An edge e between two sites in S
that can serve as a chord of any non-empty homothet of C will not be part
of DTC (S). (A homothet of C is an object that can be obtained from C
by translation and scaling; rotations are not allowed.) The subset of the
plane covered by DTC (S) is called the support hull of S (with respect
to dC ) in [288], generalizing the hull concept in Lee [493] for Lp -metrics.
The support hull is connected, as it has to contain a certain spanning tree
of DTC (S), see later, and DTC (S) is a full triangulation therein.
We remark that if the definition of DTC (S) were based on its
triangles rather than on its edges (as above), then more edges were
possibly included, dual to unbounded parallel edges of VdC (S). In
particular, the corresponding support hull could enlarge.
Flipping edges in shape Delaunay tessellations is considered in detail in
Paulini [588]. In principle, the empty shape property can be used to decide
‘Delaunayhood’ of an edge (cf. Section 4.2), but some care has to be taken.
Let e be any edge spanned by the point set S. For the given convex shape C,
consider its homothetic copy of infinite size, and denote with CL and CR
its two translates having the endpoints of e on the boundary, from the left
and the right side, respectively. (Without loss of generality, we assume that
edge e is not horizontal.) We distinguish three disjoint (open) domains in
the plane R2 as shown in Figure 7.9, defined as
I = CL ∩ CR ,
II = R2 \ (CL ∪ CR ),
III = (CL ∪ CR ) \ (CL ∩ CR ).
and
Chapter 7. General Spaces & Distances
138
II
CL
III
C
CR
I
I
e
III
II
Figure 7.9. Infinite copies of a polygonal shape C (left), and the three domains they
induce for a particular edge e (right).
Clearly, if the shape C is smooth (for example, a circular disk, as for the
classical Delaunay triangulation), then CL and CR are just the two open
halfplanes bounded by the straight line supporting e, and the distinction
above is not meaningful, because I = II = ∅, and III = R2 \ is the only
domain.
For the non-smooth case, if domain I contains any point from S, then
edge e cannot be an edge of DTC (S). Also, no point in domain II can form
a valid triangle with edge e, whereas a point p in domain III can do so,
because an (otherwise empty) homothet covering e and p might exist. In the
case where S, apart from e’s endpoints, lies entirely within the domain II ,
edge e is part of the support hull of S, from both sides. Indeed, this situation
might happen for all valid edges, and DTC (S) then degenerates to a tree,
containing no valid triangles.
It is well known that any given triangulation T of S can be transformed
into the (classical) Delaunay triangulation DT(S) of S, by applying a
sequence of edge flips based on the ‘local Delaunayhood’ of an edge (see
Section 4.2 and Subsection 6.3.3). Local Delaunayhood is defined via the
empty-circle property, but can be equivalently characterized by an angle
criterion. Let Q be any convex quadrangle in T , given as the union of
two triangles ∆1 and ∆2 in T . These triangles share an edge e. We flip e,
and consider the two triangles ∆3 and ∆4 that the quadrangle Q is split
into now. Then, if e was not locally Delaunay, the smallest angle arising
in ∆1 and ∆2 is smaller than the smallest angle arising in ∆3 and ∆4 .
That is, the angularity of T increases lexicographically. This property is,
unfortunately but not surprisingly, lost for general convex shapes C, and a
related criterion that maintains the analogy between DT(S) and DTC (S)
7.2. Convex distance functions
139
is of interest. Indeed, for DT(S) we also have the inequalities
max{r1 , r2 } < max{r3 , r4 }
and
min{r1 , r2 } < min{r3 , r4 },
where ri denotes the circumradius (radius of the circumcircle) of triangle ∆i
for i = 1, 2, 3, 4. While there are counterexamples for the min-inequality for
general convex shapes C, the max-inequality still holds [588]. Thus, each
flip in the triangulation T lexicographically reduces the sorted vector of
triangle circumradii (i.e., scaling factors of C). It also can be shown, by
adapting the proof of Theorem 4.3 in Section 4.2, that T can always be
flipped into DTC (S): Arbitrarily creating edges which are locally Delaunay
with respect to C will terminate in a unique structure. As a consequence,
DTC (S) minimizes the largest circum-shape, for all possible triangulations
of the point set S.
Note that the smallest enclosing shape of a triangle need not be unique.
Alonso et al. [48] discuss this and related issues in detail.
Lambert [482] elaborated on the question about which flip operations,
based on a given set of convex shapes, guarantee a unique locally optimal
triangulation (i.e., one where no improving flips do exist). He proved that
the underlying set of shapes has to be a homothetic family, as it is the case
for DTC (S).
The algorithmic question of how many improving flips are required
to transform a given triangulation T of S into DTC (S) remains open.
(Of course, the lower bound of Ω(n2 ) for the classical Delaunay
triangulation in Subsection 6.3.3 carries over, if C can be any convex
shape.) Of interest is also the corresponding recognition problem for shape
Delaunay tessellations: Given T , does there exist a convex shape C such
that T = DTC (S)? This problem was recently shown to be NP-hard [607].
The Delaunay triangulation is not only useful because its triangles
optimize angles and circumradii (and other quantities), but also because
it contains various subgraphs which arise in different applications. The
minimum spanning tree, the nearest neighborhood graph, and the Gabriel
graph are among these subgraphs; see Section 8.2. When considering
whether such structures extend to general convex distance functions dC , we
encounter two problems: First of all, the function dC is not a metric unless
the shape C is (point-)symmetric. Also, dC depends on the chosen center v
of C, whereas the triangulation DTC (S) does not. Still, a convex distance
function can be constructed that is center-independent and symmetric, and
which yields the desired subgraph property for minimum spanning trees and
Gabriel graphs [588], as we describe next.
140
Chapter 7. General Spaces & Distances
Figure 7.10. DTC (S) for triangle
shape C contains the minimum
spanning tree for the metric mC .
Figure 7.11. DTĈ (S) for Minkowski
sum shape Ĉ contains the same
minimum spanning tree (from [588]).
Assume that C is centered at the origin, and denote with Cv , for v ∈ C,
the translation of C by v. Given two points p and q, we define the distance
function
mC (p, q) = min dCv (p, q).
v∈C
Observe that mC (p, q) equals the Euclidean distance d(p, q), divided by
the length of the longest line segment contained in C and parallel to
the edge pq. Thus the concept of the, in general, non-symmetric radius
(see the beginning of Subsection 7.2.1) is replaced by the concept of
symmetric diameter. That is, mC (p, q) is just the scaling factor of the
smallest homothet of C that covers pq. To see that we really deal with
a convex distance function, observe further that
Cv .
mC (p, q) = dĈ (p, q), with Ĉ =
v∈C
The symmetric convex shape Ĉ is just the Minkowski sum of C and its
reflected shape. (The Minkowski sum of two sets A and B in the plane is
defined as A ⊕ B = {a + b | a ∈ A, b ∈ B}. Loosely speaking, A ⊕ B is the
union of all translates of B that overlap with A.)
Coming back to the question of subgraphs, we can now use the following
facts: For any symmetric shape, and in particular for Ĉ, the equality
mĈ (p, q) = 12 dĈ (p, q) holds, such that the two metrics mĈ and dĈ yield
the same minimum spanning tree. Further, we have mĈ (p, q) = 12 mC (p, q),
for general convex shapes C. In conclusion, the minimum spanning tree
with respect to the metric mĈ , which is contained in DTĈ (S) because mĈ
is symmetric, is the minimum spanning tree of S with respect to mC . This
tree is indeed a subgraph of the original shape Delaunay triangulation,
DTC (S), as can be easily proven by using the empty-shape property
of DTC (S). Figures 7.10 and 7.11 illustrate these facts. Notice that DTC (S)
7.2. Convex distance functions
141
and DTĈ (S) can be quite different triangulations, even concerning their
support hulls.
Similar subgraph properties also hold for the Gabriel graph of S
(a supergraph of the minimum spanning tree), because this graph is defined
by emptiness of the smallest homothet, Cpq , of C that covers an edge pq.
Interestingly, the Gabriel graph coincides with DTC (S) for any triangular
shape C, because then the ‘forbidden’ domain I for an edge pq (compare
Figure 7.9) just is Cpq .
7.2.3. Situation in 3-space
The definition of a convex distance function, and in particular of the
Lp -norms, can easily be extended to dimensions higher than 2. In this
subsection we study convex distance functions in 3-space. Such a distance
function is based on a convex, compact set C in 3-space which contains the
origin in its interior. We call such a unit sphere C good if it is, in addition,
strictly convex, symmetric, and smooth.
Some of the pleasant properties of two-dimensional convex distance
functions still have their counterparts in 3-space. For example, the bisector
B(p, q) of two points is a surface homeomorphic to the plane, and two such
bisector surfaces B(p, q), B(p, r) have an intersection homeomorphic to the
line, or empty. In these aspects, convex distance functions in 3-space do
not differ from the Euclidean distance. But whereas a Euclidean sphere
is uniquely determined by four points in space, this is no longer true for
convex distance functions. In Icking et al. [418] the following result has been
shown; consult Figure 7.12.
Theorem 7.3. For each n > 0 there exist a good convex set C and
four points in 3-space, such that there are 2n + 1 homothetic copies of C
containing these points in their boundaries. This number does not decrease
when the four points are independently moved within small neighborhoods.
The center v of a scaled and translated copy of C containing p, q, r, s in
its boundary is of the same dC -distance to each of these points. Hence, v is
a Voronoi vertex in the diagram VdC ({p, q, r, s}). Therefore, Theorem 7.3
implies that there exists no global upper bound to the number of vertices
of the Voronoi diagram of four points in 3-space, that holds for arbitrary
convex distance functions.
If it is known that for a particular convex distance function dC no more
than k homothetic copies of C can pass through any four points in general
position, then, obviously, the complexity of the Voronoi diagram of n points
142
Chapter 7. General Spaces & Distances
Figure 7.12. As vertices r and p move upward along the convex curve f , while the
tetrahedron T defined by {r, p, q, s} stays homothetic, vertices q and s write out a convex
curve, g. The convex hull of f and g forms the lower half of a convex body C. Since C
touches the vertices of infinitely many homothetic copies of T , an infinite number of
homothetic copies of C are passing through {r, p, q, s}. This construction can be tuned
to yield a proof of Theorem 7.3.
based on dC is in O(kn4 ), because at least four Voronoi regions meet at a
Voronoi vertex.
Lê [487] has obtained the following result on the number k for the family
of Lp -norms, using the fact that they are defined by algebraic equations of
degree p. The proof uses results from the theory of additive complexity, see
Benedetti and Risler [120]. This and the subsequent results require that the
sites be in general position.
Theorem 7.4. There exists an upper bound to the number of Lp -spheres
in d-space that can pass through d + 1 given points, which depends only on d
but not on p.
The fourth power of n in the general upper bound of O(kn4 ) for Voronoi
diagrams in dimension 3 is believed to be far off. For semi-algebraic distance
measures of constant degree, a general result by Sharir [639] on lower
envelopes implies an upper complexity bound of only O(nd+ε ). This result
applies to the Lp family, but the constant in O tends to ∞ as p grows.
For the three-dimensional Euclidean Voronoi diagram, the true worstcase complexity is only Θ(n2 ); see Section 6.1. Improving on previous
work by Boissonnat et al. [147] and Tagansky [673], Icking and Ma [417]
were able to generalize this tight bound to polyhedral convex distance
functions.
7.2. Convex distance functions
143
Theorem 7.5. Let C be a fixed convex polyhedron in 3-space. Then the
Voronoi diagram with respect to dC of n points is of complexity O(n2 ).
The constant in O is proportional to k 3 α(k), where k denotes the
complexity of C, and α denotes the (extremely slowly growing) inverse
of Ackermann’s function. The proof makes use of the fact that such edges
of bisector surfaces, that are generated by an edge of C, can be of at most
k different slopes.
Boissonnat et al. [147] have also shown that the Voronoi diagrams of
n points in d-space based on the L∞ -norm, or on a simplex as the unit
sphere, are both of complexity Θ(nd/2 ).
The unit spheres for the L∞ -norm and the L1 -norm are hypercubes
and cross-polytopes (i.e., duals of hypercubes), respectively. (A hypercube
in d-space is the Cartesian product of d mutually orthogonal line segments
of equal length; for example, a square in the plane, or a cube in 3-space
where the dual cross-polytope is the regular octahedron.) In contrast
to hypercubes, cross-polytopes have a large number of facets in high
dimensions, which partially explains why tight bounds for L1 -diagrams are
only available for dimensions up to three.
For the complexity of the Voronoi diagram for lines in 3-space under
the Euclidean distance only an O(n3+ε ) upper bound is known [639] (unless
the number of orientations is bounded [470]). Confirming the impression
that polyhedral distance functions are somewhat simpler than the Euclidean
distance, Chew et al. [217] were able to provide a smaller bound for lines
under a polyhedral convex distance function.
Theorem 7.6. Let C denote a convex polyhedron of constant complexity
in 3-space. Then the Voronoi diagram of n lines based on dC is of complexity
O(n2 α(n) log n). A lower bound is given by Ω(n2 α(n)).
Koltun and Sharir [469] have generalized Theorem 7.6 to line segments.
For more general sites, they obtained the following.
Theorem 7.7. Let C denote a convex polyhedron of constant complexity
in 3-space. Then the Voronoi diagram of polyhedral sites of total complexity
n has a size of O(n2+ ).
This includes the practically relevant case of the medial axis of a
(nonconvex) polyhedron P in 3-space, under a convex polyhedral distance
function. If some point in the ‘unit ball’ C is fixed as its center, then the
medial axis is the union of the centers of all homothets of C that can be
144
Chapter 7. General Spaces & Distances
inscribed in P . Defined this way, the medial axis is a piecewise linear set,
which can be constructed efficiently; see Aichholzer et al. [26].
The reader interested in purely mathematical properties of convex
distance functions is referred to the survey by Martini et al. [517, 518]
and to the monograph by Thompson [684].
7.3. Nice metrics
Convex distance functions do not apply to environments where the distance
between two points can change under translations. This happens in the
presence of obstacles, or in cities whose streets do not form a regular grid.
In order to model such environments in a more realistic way, we can use
the concept of a metric.
In this section we consider point sites and Voronoi diagrams that are
based on a metric, m, in the plane. (However, a metric can be defined
in arbitrary spaces). It associates with any two points, p and q, a nonnegative real number m(p, q) which equals 0 if and only if p = q holds.
Moreover, we have symmetry m(p, q) = m(q, p), and the triangle inequality
m(p, r) ≤ m(p, q) + m(q, r). The set
{z ∈ R2 | m(a, z) < r}
will be called an (open) m-disk of radius r centered at point a.
General metrics are quite a powerful modeling tool. Suppose there is
an air-lift between two points a and b in the plane. Then

 d(p, q),
m(p, q) = min d(p, a) + f + d(b, q),

d(p, b) + f + d(a, q),
describes how fast one can travel from p to q; we assume that going by car
takes time equal to the Euclidean distance, d, whereas the flight between
a and b takes time f < d(a, b). It is easy to verify that the function m is a
metric in the plane: We could warp the plane so that a and b have Euclidean
distance f in 3-space, and glue on a straight handle connecting them; the
function m describes the lengths of shortest paths in the resulting space.
7.3.1. The concept
Voronoi regions with respect to the air-lift metric m need not be connected.
Suppose site q lies on the line segment ab, as shown in Figure 7.13. In
order to construct the Voronoi diagram of the two sites a and q we use the
expanding waves approach mentioned in Chapter 2, and let two Euclidean
7.3. Nice metrics
145
r
r-f
q
a
Dm(a,q)
b
Dm(q,a)
B(a,q)
Dm(a,q)
H
Figure 7.13. If there is an air line connecting a and b, the Voronoi region of a with
respect to q is disconnected.
circles expand from a and q at the same speed. Their intersections form
the Euclidean bisector B(a, q). As soon as the radius of these circles has
reached f , the duration of the flight from a to b, a new circle starts growing
from b. Subsequently, its radius is always by f smaller than the radius r
of the other two circles. The intersections of the circles expanding from q
and from b form a hyperbola, H; it is the locus of all points x satisfying
d(q, x) − f = d(b, x).
Such Voronoi diagrams of n points with additive weights are also known
as the Voronoi diagram for circles. They often appear quite unexpectedly.
One can construct them in time O(n log n) by different algorithms; see
Section 5.5, and especially Subsection 7.4.1.
Clearly, the region of a with respect to air-lift metric m, Dm (a, q),
consists of the two parts shaded in Figure 7.13; they are not connected.
In order to exclude such phenomena we restrict ourselves to a subclass of
metrics introduced in Klein and Wood [467]. First, we define the type of
bisector curve we would like to work with.
Definition 7.1. A curve J in the plane is called nice if its stereographic
projection to the sphere can be completed to a Jordan curve through the
north pole. (A Jordan curve on the sphere, or in the plane, is a closed curve
without self-intersections.)
A nice curve J in the plane is simple (i.e., no self-intersections),
unbounded, and homeomorphic to a straight line. It splits the plane into
two unbounded, open domains, D1 and D2 . Both of its ends tend to infinity.
Just as each point p on J can be accessed from Di , i = 1, 2, by an arc that
146
Chapter 7. General Spaces & Distances
runs entirely in Di but for its endpoint p, the same holds for the north pole.
Any straight line, parabola, or hyperbola is nice, but a circle is not.
Definition 7.2. A metric m in the plane is called nice if it enjoys the
following properties:
(1) Each m-disk is contained in a Euclidean disk, and vice versa; each
m-disk contains a Euclidean disk, and vice versa.
(2) For any two points p, r, there exists a point q different from p and r
such that m(p, r) = m(p, q) + m(q, r) holds.
(3) For any two points p, q, the boundary of the bisector Bm (p, q) consists
of two nice curves (that may partially or fully coincide).
In the sequel we shall discuss some implications of these properties that
will be relevant for Voronoi diagrams later on.
Property (1) of Definition 7.2 ensures that ‘close’ and ‘far’ mean the
same under m as in the Euclidean metric d, only the quantification of how
close or far may differ. The open sets are the same for either metric.
Moreover, the plane endowed with the metric m is complete, in the
following sense. If a sequence (ai )i>0 fulfills the Cauchy condition under
metric m (that is, if for each ε > 0 there is an index threshold N such that
m(ai , aj ) < ε holds for all i, j ≥ N ), then a limit point a∗ exists in the
plane to which the sequence converges.
Property (2) implies that one can always find a point q different from
p, r that makes the triangle inequality
m(p, r) ≤ m(p, q) + m(q, r)
an equality. Such a point q is said to lie ‘between’ p and r. Once q is found,
one can find points in between p and q, and between q and r, and so on.
Thanks to the completeness guaranteed by (1), these points form, in the
limit, a path π from p to r, which has a remarkable property.
Definition 7.3. A path π is called m-straight if for any three consecutive
points a, b, c on π the equality m(a, c) = m(a, b) + m(b, c) holds.
Theorem 7.8. (Menger [541]) Let m be a complete metric space that
fulfills Property (2) of Definition 7.2. Then for any two points p, r there
exists an m-straight path connecting them.
The following observation shows why m-straight paths are useful. As in
the standard definition of arc length, one can put many consecutive points
on a path π and add up their m-distances. If a limit exists, for arbitrarily
fine resolution, it is called the m-length of π. For an m-straight path from p
7.3. Nice metrics
147
to r, its m-length equals m(p, r), as a direct consequence of Definition 7.3.
No other path from p to r can have a smaller m-length. Thus, m-straight
paths are shortest paths under the metric m, the true analogue of straight
segments in the Euclidean metric.
Some authors call m-straight paths geodesics, others use this name for
the wider class of paths that cannot be shortened locally.
Property (3) of Definition 7.2 refers to the metric bisector
Bm (p, q) = {z ∈ R2 | m(p, z) = m(z, q)}.
It may not be a nice curve itself, as the example depicted in Figure 7.4(iii)
shows. In this case, one of the two boundaries of Bm (p, q) will be chosen
as a bisecting curve, and we want this boundary to be a nice curve; see
Definition 7.5 below.
Lemma 7.2. Each convex distance function is nice.
Proof. Let dC be a convex distance function based on a convex set C.
Then Property (1) of Definition 7.2 is obviously fulfilled, as the dC -disks
are scaled copies of C. Property (3) follows from the discussion of Figure 7.5.
Property (2) is also guaranteed; each interior point q of the line segment
connecting p with r lies in between p and r.
More generally, dC -straight paths of convex distance functions can be
described as follows. Suppose that the halfline from p to r cuts the boundary
of Cp , the translated copy of C centered at p, at some point c. If c is interior
point of a straight segment vw in the boundary, then exactly the v- and
w-monotone paths from p to r are dC -straight. If c is not an interior edge
point, the line segment pr is the only dC -straight path from p to r; see
Figure 7.14.
A statement converse to Lemma 7.2 also holds. If a nice metric m is
invariant under translations, and if m(p, r) = m(p, q) + m(q, r) holds for
Figure 7.14.
A dC -straight path in a convex distance function.
Chapter 7. General Spaces & Distances
148
any three consecutive points p, q, r on a line, then m is a convex distance
function.
More complex examples of nice metrics are composite metrics that
result from assigning different convex distance functions to the regions of a
planar subdivision, or the Karlsruhe (or Moscow ) metric; see Klein [461].
Here a center point, 0, is fixed and only such paths are allowed that consist
of pieces of circles around 0 and of pieces of radii from 0. The Karlsruhe
metric m(p, q) denotes the minimum Euclidean length of an allowed path
from p to q.
It depends on the angle between p and q if the shortest connecting
path runs through the center, or around it; see Figure 7.15(i). The
bisector of two points consists of up to eight segments that are pieces of
straight lines or hyperbolas if written in polar coordinates. In (ii), the
Voronoi diagram of eight point sites under the Karlsruhe metric is
shown.
Clearly, in the example of the air-lift metric considered earlier,
Property (2) of Definition 7.2 is violated: There is no point q satisfying
m(a, b) = m(a, q) + m(q, b) different from one of the two airports, a and b.
(The points in mid-air do not count, as they are not in R2 .)
7.3.2. Very nice metrics
Lemma 7.2 in the preceding subsection can be sharpened in the
following way.
q2
0
0
114,59
q1
p
(i)
(ii)
Figure 7.15. The shortest path in the Karlsruhe metric runs through the center iff the
angle between p and q exceeds about 114, 59◦ . In (ii) a Voronoi diagram in the Karlsruhe
metric is shown.
7.3. Nice metrics
149
Definition 7.4. A nice metric m is called very nice if, for any three points
a, b, c, one of the following properties is fulfilled.
(1) Each m-straight path from a to b contains c.
(2) Each m-straight path from a to c contains b.
(3) There exist m-straight paths πb from a to b, and πc from a to c, that
have only point a in common.
Definition 7.4 clearly applies to the Euclidean metric; a, b, and c are
collinear, or the line segments from a to b and c are disjoint, except for
point a. From the above characterization of dC -straight paths, the following
generalization is immediate.
Lemma 7.3. Each convex distance function is very nice.
The Karlsruhe metric, however, is nice, but not very nice. All straight
paths from a point p to points in a wedge ‘behind’ the center, O, are passing
through O, but they do not contain each other.
Now we define the Voronoi diagram Vm (S) based on a nice metric m.
Let ‘≺’ be some order relation on S, for example, the lexicographical order.
Definition 7.5. (i) For p, q ∈ S, let
Cm (p, q) = {z ∈ R2 | m(p, z) ≤ m(z, q)} if p ≺ q
Rm (p, q) =
Dm (p, q) = {z ∈ R2 | m(p, z) < m(z, q)} if q ≺ p
denote the halfplane associated with p. Let Rm (q, p) be its complement,
the halfplane associated with q.
(ii) Define
Rm (p, q)
VRm (p, S) =
q∈S\{p}
to be the Voronoi region of p with respect to S under m, and let
Vm (S) =
VRm (p, S) ∩ VRm (q, S)
p,q∈S, p=q
denote the Voronoi diagram of S under m. (Here A denotes the closure of
a set A, that is, the union of A with its boundary.)
By this definition, the border J(p, q) between the halfplanes of p and q
equals the common boundary of Bm (p, q) with Dm (q, p), if p ≺ q, and the
common boundary of Bm (p, q) and Dm (p, q), if q ≺ p. By Definition 7.2(3),
both of these boundaries are nice curves.
150
Chapter 7. General Spaces & Distances
Definition 7.5 puts each Voronoi edge in the region of the smaller of the
two sites it separates. An advantage of this technical modification becomes
apparent in Lemma 7.4.
Let us generalize the standard notion of star-shapedness to general
metrics.
Definition 7.6. A set V is called m-star-shaped as seen from point p in V
if, for each point x ∈ V , each m-straight path from p to x is fully contained
in V .
Lemma 7.4. Let m be a nice metric. Then each Voronoi region VRm (p, S)
is m-star-shaped as seen from its site p, hence pathwise connected.
Proof. Since m-star-shapedness is stable in intersections, we need only
prove it for each set Rm (p, q), according to Definition 7.5. Suppose that
Rm (p, q) = Cm (p, q) holds. Let π be an m-straight path from p to some
point x ∈ Cm (p, q), and suppose that some point y on π does not belong to
Cm (p, q). That is, we have m(p, x) ≤ m(x, q) but m(p, y) > m(q, y). By the
straightness of π we obtain
m(p, x) = m(p, y) + m(y, x)
> m(q, y) + m(y, x) ≥ m(q, x),
which is a contradiction. Hence, π ⊂ Cm (p, q). The case where Rm (p, q) =
Dm (p, q) is analogous. Theorem 7.8 ensures that for each x ∈ VRm (p, S)
an m-straight path from p to x exists. Together, these facts imply that
VRm (p, S) is a pathwise connected set.
Regions that are path-connected are often just called connected. We are
using either term, as appropriate from the context. Observe that pathconnectedness is weaker than simple connectedness: all connecting paths
might run around some ‘holes’ in the region. (To give a formal definition,
a set M in the plane is called simply connected if every Jordan curve J ⊂ M
can be contracted to a point in M .)
A Voronoi diagram all of whose regions are connected is sometimes
referred to as an orphan-free Voronoi diagram.
Sometimes it is just a single point that makes a Voronoi region
connected. An example for such a cut-point in the Karlsruhe metric is
shown in Figure 7.16. Since p, q, r have the same distance from the center,
bisector B(p, q) consists of the halfline that emanates from O and separates
p, q, and of a 2-dimensional wedge ‘on the other side’ of center O, and the
same holds for bisector B(q, r). If tie-break order q ≺ p ≺ r is chosen,
7.3. Nice metrics
151
p q
q r
VRK (q, S )
O
p
r
q
p q
Figure 7.16.
qr
The region of q has a cut-point in the Karlsruhe metric if q ≺ p ≺ r.
0
p
y
q
x
Figure 7.17. The unique shortest path between x and y in the Karlsruhe metric does
not stay within the Voronoi region of q.
the darkly shaded Voronoi region VR(q, S), where S = {p, q, r} results,
according to Definition 7.5.
Whereas the star-shapedness of the Voronoi regions is preserved in nice
metrics, the example of the Karlsruhe metric m shows that the convexity
of the Euclidean Voronoi regions has no counterpart. In Figure 7.17, the
Voronoi diagram of four point sites on the same radius is shown. There
exists exactly one m-straight path connecting the points x, y of the Voronoi
region of q, but this path is not contained in the region of q.
That the Voronoi regions under a nice metric m are m-star-shaped is a
property strong enough for computing the Voronoi diagram efficiently. The
following has been shown by Dehne and Klein [248].
152
Chapter 7. General Spaces & Distances
Theorem 7.9. The Voronoi diagram of n point sites under a nice metric
in the plane can be constructed within O(n log n) many steps, using the
sweep line approach.
It is also possible to apply the randomized incremental construction
method introduced in Section 3.2, or the divide & conquer technique.
With the latter a new difficulty arises: If we use a line to subdivide S
into subsets L and R, then the merge chain Bm (L, R) of Voronoi edges
separating L-regions from R-regions can contain cycles that would not be
detected during the merge phase.
In [461] two criteria have been introduced for Bm (L, R) to be acyclic.
For example, if the m-circles are simply connected we can use, as a divider
of S, any curve homeomorphic to the line whose intersection with all
m-circles is connected (possibly empty). This generalizes a result by Chew
and Drysdale [215] on convex distance functions, where each straight line
possesses this property.
7.4. Weighted distance functions
In this section we shall discuss a variant of distance measures where sites
can be assigned individual weights in different ways. Intuitively speaking,
the weight of a site expresses its capability to influence its neighborhood.
A prominent representative we have already encountered in Section 6.2: the
power function d(x, p)2 − w(p) of a site p with weight w(p). The resulting
weighted Voronoi diagram, the power diagram, is an important and versatile
concept that unifies and relates to many other geometric structures and
applications; most of Chapter 6 is devoted to them.
7.4.1. Additive weights
Also other types of weighted Voronoi diagrams appear, sometimes quite
unexpectedly, in many applications. As an introductory example, consider
a polygonal chain T = (p1 , p2 , . . . , pn ) on n vertices in the plane, modeling
a railroad track. Its dilation,
δ(T ) =
max
1≤i<j≤n
d(Tij )
,
d(pi , pj )
measures the maximum detour encountered when using the train instead
of travelling as the crow flies. Here, d(Tij ) denotes the Euclidean length of
the subchain (pi , pi+1 , . . . , pj ) of T .
7.4. Weighted distance functions
153
Cj
Ci
p∗j
ai + d(pi, pj )
aj
p∗i
ai
pj
Figure 7.18.
pi
An arrangement of two cones.
Suppose we want to test if δ(T ) ≤ κ holds, for some threshold value
κ > 1. This means, for each i we want to know if
κ≥
d(T1j ) − d(T1i )
d(Tij )
=
d(pi , pj )
d(pi , pj )
holds, or equivalently,
d(pi , pj ) + ai ≥ aj
with am =
d(T1m )
κ
(7.1)
for each j > i. This formula has an interesting geometric interpretation.
Let Cm denote the upright cone of angle 90◦ in 3-space whose apex p∗m is
positioned at height am above point pm in the xy-plane; see Figure 7.18.
Clearly, d(pi , pj ) + ai ≥ aj holds if, and only if, point p∗j lies below cone Ci .
Since ai ≥ aj whenever i ≥ j, we have δ(T ) ≤ κ if and only if each apex p∗j
appears on the lower envelope of the cones Cj .
In Section 3.5 we have seen how Voronoi diagrams can be obtained from
lower envelopes of so-called distance cones. One can imagine that from each
site p a circle expands at unit speed, forming a 90◦ cone Cp in 3-space. The
lower envelope of the arrangement of these cones, when projected vertically
to the xy-plane, equals the Voronoi diagram of the sites p ∈ S.
The above dilation problem leads us to study more general
arrangements, where each cone Cp has been lifted up by a certain
distance, ap , above the xy-plane, modeling that expansion of circles start
at individual times. We may assign to each site p the weighted distance
function d(p, z) + ap , which describes the vertical distance from z ∈ R2 to
cone Cp . The projection of the lower envelope of the arrangement of all
cones Cp , where p ∈ S, equals the Voronoi diagram of S under these distance
functions.
154
Chapter 7. General Spaces & Distances
4
5
7
1
2
0
4
Figure 7.19. Additively weighted Voronoi diagram for seven sites. Numbers express site
weights (radii) wp = −ap . (Dashed circles σp are intersections of the cones Cp with the
xy-plane.) A ‘hot’ real-world interpretation is that grass fires starting at locations p at
times ap meet in the hyperbolic region borders (Calabi and Hartnett [176]). In a ‘cool’
interpretation the edges of the diagram are viewed as the interference pattern of waves
emanating from pebbles p tossed at times ap into a quiet pond.
The same structure also appears as the Voronoi diagram for circles in
the Euclidean metric. A point z ∈ R2 outside a circle σp with radius wp
around site p has a distance of
dp (z) = d(p, z) − wp
to σp . If z lies inside σp we consider its distance to σp as negative. Then,
the Voronoi diagram of the circles σp is the additively weighted Voronoi
diagram of S, with site weights wp .
Figure 7.19 depicts an instance of this diagram. Time-shifted ‘wave’
expansions are indicated by arrays of co-centric circles, with outermost
circles σp .
The bisector of two weighted sites p and q is a hyperbola bent towards
p if wp < wq , or a straight line if wp = wq . The Voronoi region of p is empty
if σp fully lies inside some other circle σq . All Voronoi regions are simply
connected, in fact, star-shaped as seen from the respective sites, by the same
argument as in Lemma 7.4. Hence, the additively weighted Voronoi diagram
is of linear complexity; cf. Lemma 2.3. Note, however, that two regions in
7.4. Weighted distance functions
155
this diagram may be adjacent in as many as O(n) edges — for example
if many sites of small weight are placed near the bisector of two heavily
weighted sites.
The additively weighted Voronoi diagram is also known as Johnson–
Mehl model in mineralogy [433]. In computational geometry, it has been
studied, e.g., by Lee and Drysdale [496] and Sharir [638]. Fortune [344] has
provided a construction algorithm that runs in time O(n log n), using the
sweep-line approach (Section 3.4). Also, the divide & conquer algorithm
in [25] (Section 5.5) applies, which can handle arbitrary circular arcs.
Thus, one can solve the aforementioned dilation problem in time
O(n log n). We just construct the additively weighted Voronoi diagram of
the chain’s vertices pi , equipped with the weights wi = −ai of Formula 7.1,
and check if each pi has a non-empty Voronoi region; see Agarwal et al. [14]
and Grüne et al. [385] for further results on the dilation problem.
We have already encountered the additively weighted Voronoi diagram
in the context of air-lift metrics in Section 7.3. Another surprising
application is in the Monge–Kantorovich transportation problem; see
Gangbo and McCann [350] and Caffarelli et al. [175]. Imagine an area
A with a set of n construction sites, pi . Area A is covered with sand,
according to some Lebesgue-continuous probability distribution (the
uniform distribution, for example). Each site has a demand for a share
of σi > 0 of sand, where i σi = 1. How should sand be transported to
sites, in order to minimize transportation cost?
If one assumes that the cost of moving a unit of sand from z to p is
given by a strictly convex function h(p − z), then the optimum solution
equals the projection of the lower envelope of an arrangement of h-cones
(distance cones)
Cp = {(z, h(p − z)) | z ∈ R2 } + ap .
In general, the additive values ap must be determined by numerical
methods.
A simple proof for rather general distance measures can be found in
Geiß et al. [358]. If h(z) = |z|2 , then a power diagram represents the
optimum solution; see Aurenhammer et al. [97] and Section 6.4. Related
is the minimum-cost load balancing problem studied in Aronov et al. [69],
who show that the respective optimal partition of a convex planar domain
is induced by an additively weighted Voronoi diagram.
To see one more occurrence of this diagram, imagine some tilted plane T
in R3 , passing through the origin O. As T is not horizontal, distances on T
may be defined as the Euclidean distance plus a multiple k > 0 of the signed
156
Chapter 7. General Spaces & Distances
difference in height. The cost of ‘moving’ on T thus depends not only on
the Euclidean distance but also on how much upwards or downwards the
movement has to travel, simulating the situation when driving a vehicle
on T . This so-called skew distance is direction-sensitive and, in particular,
non-symmetric, and is investigated in Aichholzer et al. [31]. Its ‘unit
circle’ C is a conic with focus O, directrix the horizontal line y = 1/k,
and eccentricity k. Thus C is an ellipse for 0 < k < 1, and unbounded
otherwise, namely a parabola for k = 1, and a branch of a hyperbola
for k > 1.
Now consider a set S of n point sites on T , without loss of generality
in its y-positive halfplane. To each site p ∈ S, assign the value k · py as its
weight (py > 0 denotes the y-coordinate of p). Then the additively weighted
Voronoi diagram for these sites coincides with the Voronoi diagram for the
skew distance on T . As a consequence, this so-called skew Voronoi diagram
can be constructed in O(n log n) time and O(n) space.
A similar concept is the so-called boat-sail distance in Sugihara [667],
whose definition underlies a given flow field (of a river, for example).
Its Voronoi diagram can be computed by a plane sweep (Section 3.4) in
O(n log n) time for homogeneous flow fields of any speed, but tends to
get quite complex for anisotropic flows, where approximate construction
methods exist; see Nishida and Sugihara [565].
7.4.2. Multiplicative weights
In the circle expansion model explained before, it is natural to ask what
happens if circles expand from their sites at different speeds. If we assign
to each site p a multiplicatively weighted distance function
dp (z) =
d(p, z)
wp
where wp > 0, then sites of larger weights emit circles that expand more
quickly, resulting in distance cones
Cp = {(z, dp (z)) | z ∈ R2 }
of larger opening angles. This has quite drastical consequences for the
multiplicatively weighted Voronoi diagram, which results from projecting
the lower envelope of the cones to the plane. We will discuss this in the
following.
Since flat cones are always penetrated by steeper cones, the bisector of
two sites must be a closed curve, circumscribing the site of lesser weight.
Apollonius [64] discovered the following.
7.4. Weighted distance functions
157
Lemma 7.5. The locus
B(p, q, r) =
z ∈ R2
d(p, z)
=r
d(q, z)
of all points z in the plane, whose Euclidean distances to p and q have a
fixed ratio r = 1, is a circle, later named Apollonius circle after him.
Proof. Apollonius knew that an inner angular bisector divides the opposite
edge of a triangle to the proportion of its adjacent edges. Thus, in
Figure 7.20
d(p, z)
d(p, e)
=
=r
d(e, q)
d(z, q)
holds, and analogously
d(p, z)
d(p, e )
=
=r
d(q, e )
d(z, q)
is fulfilled for the outer angular bisector ze of the triangle (p, q, z).
By construction, ze and ze form a right angle. Thus, by Thales’ theorem
(which asserts that every angle in the half-circle is a right angle), there
Figure 7.20. An Apollonius circle bisecting the multiplicatively weighted sites p and q
in the plane.
Chapter 7. General Spaces & Distances
158
exists a circle passing through e, e and z. From
d(p, e) =
1
r
d(p, q) and d(e, q) =
d(p, q)
r+1
r+1
together with
r=
d(p, q) + d(q, e )
d(p, q)
d(p, e )
=
=
+ 1,
d(q, e )
d(q, e )
d(q, e )
the diameter and center of this circle can be computed to
2r
d(p, q),
−1
1
d(q, c) = d(e, c) − d(e, q) = 2
d(p, q).
r −1
d(e, e ) = d(e, q) + d(q, e ) =
r2
It follows that this circle, C(r), depends only on p, q, and ratio r, but
not on the particular point z ∈ B(p, q, r). Therefore, B(p, q, r) ⊆ C(r).
Since circles C(r) and C(r ) are disjoint if r = r , B(p, q, r) = C(r) follows,
for each r = 1.
With circular bisectors, Voronoi regions are no longer simply connected.
Even worse, regions can become disconnected, causing the complexity of the
Voronoi diagram to grow. (One should observe that the proof of Lemma 7.4
does not extend to multiplicative weights.)
In [96], Aurenhammer and Edelsbrunner have investigated the
multiplicatively weighted Voronoi diagram of n points, which is also referred
to as the Apollonius model. They proved that its worst-case complexity
is Θ(n2 ); an example is sketched in Figure 7.21. Also, an optimal O(n2 )
algorithm for constructing this diagram is provided, based on its embedding
in power diagrams in 3-space. (Interestingly, also the additively weighted
Voronoi diagram enjoys such an embedding; cf. the end of Section 6.2.)
Statistical data on the behavior of diagrams in the multiplicative case can
be found in Sakamoto and Takagi [615].
Multiplicatively weighted sites are suited to model facilities of different
attractiveness to customers, such as shops or service stations like radio
senders; see [96] and references therein. It is instructive to visualize the
effect of disconnected Voronoi regions in such real-world scenarios. For
example, a customer located close to both a small shop and a supermarket
will usually choose to visit the bigger, more attractive shop. When getting
sufficiently close to the small shop, it may get his/her attention, though.
At last, the situation reverses again when both shops are further away.
Similarly, when driving in a car, the same (strong) radio sender will be
dominantly received several times, after having passed by weaker senders.
7.4. Weighted distance functions
159
40
10
40
40
40
20
20
7
Figure 7.21. A Voronoi diagram of n point sites with multiplicative weights can be of
complexity Θ(n2 ). Numbers indicate site weights.
Diverse applications exist, in economy, geographical information systems,
and physics; see Gambini et al. [349], Boots [152], and Reitsma et al. [604]
who also mention different areas of use.
In biology, Bock et al. [135] used multiplicatively weighted Voronoi
diagrams in order to describe the dynamic behavior of cell tissue. They
model inner cell bodies as disks of radius wi , and consider the weighted
Voronoi diagram of the disks’ centers, pi , with multiplicative weights wi .
In order to avoid disconnected cells, the Voronoi region of pi is restricted
to a Euclidean disk of radius wi · R, for some constant R > 1. While this
modification causes all regions to be bounded, there may now be ‘holes’
in the cell structure, that is, areas that do not belong to any region.
Interestingly, the following property holds.
Lemma 7.6. Let wmin and wmax denote the smallest and largest
multiplicative weight, respectively. If
R≤
wmax + wmin
wmax − wmin
holds for the restriction radius R, then all restricted Voronoi regions are
star-shaped, as seen from their sites pi , hence connected. That is, the
diagram becomes orphan-free.
In a more realistic model, cells are represented by ellipses rather than
circles; see Bock [136]. This approach leads to anisotropic Voronoi diagrams
that will be introduced in Subsection 7.4.4.
Multiplicatively weighted Voronoi diagrams also arise as special cases
of a structure called angular Voronoi diagram. Given a set of n sites of
general shape (one-dimensional or two-dimensional objects), the plane is
divided into regions from where a particular site is seen under the largest
angle, compared to the remaining sites. If the sites are disks of various
radii, a multiplicatively weighted Voronoi diagram results. In fact, the locus
of all points x that ‘see’ two given disks at the same angle is a circle.
The case of line segment sites is treated in Asano et al. [78]. The best
known upper bound is now only O(n2+ε ) instead of O(n2 ). Applications to
160
Chapter 7. General Spaces & Distances
mesh improvement are plausible, based on finding vertex placements that
minimize the largest visual angle for the surrounding edges.
Angular Voronoi diagrams are quite different from the visibilityconstrained Voronoi diagram (Section 5.4) or the polar Voronoi diagram
(Subsection 8.5.1).
7.4.3. Modifications
Adapting to practical needs — not least for more realistic modeling of
natural phenomena — weighting schemes have been modified, sometimes
(but not always) with drastical changes of the resulting weighted Voronoi
diagram.
As a ‘harmless’ modification, other than the Euclidean metric can be
used. For instance, Lee and Torpelund-Bruin [501] apply multiplicatively
weighted Voronoi diagrams for the Lp -metrics in the design of decision
support systems.
Of interest are also farthest-site weighted Voronoi diagrams, where
each site is associated with the region at farthest (rather than at closest)
weighted distance; cf. Subsection 6.5.1. In fact, the geometric properties
of the diagrams remain unchanged, because they are still projections
of envelopes of cones — now the upper envelope has to be taken.
Combinatorially, a big difference may occur in the multiplicative case, as
has been shown in Lee and Wu [500]: the worst-case size reduces to O(n).
A similar reduction in size can be observed for planar Voronoi diagrams
in the L1 -metric (Section 7.2; from O(n) to constant size), and for the
Voronoi diagram for crossing line segments (Section 5.1; from O(n2 ) to
O(n) size). On the other hand, power diagrams and additively weighted
Voronoi diagrams retain a linear worst-case size in their farthest-site
variants.
The weighting schemes discussed in the preceding subsections are
obviously combinable. For example, a mixture of the power function and
the multiplicatively weighted distance, sometimes called a mixed weighted
distance,
∆(x, p) =
d(x, p)2 − w1 (p)
w2 (p)
arises in the context of clustering as a certain intra-cluster measure; see
Inaba et al. [421], and Section 8.4 for geometric clustering methods related
to Voronoi diagrams.
A mixed weighted distance of the same type also applies to the following
graph drawing problem. Eppstein [324] recently proved that every planar
7.4. Weighted distance functions
161
graph with maximum degree 3 has a planar drawing in which the edges
are circular arcs that meet at equal angles around every vertex. Such
drawings are called Lombardi drawings and have perfect angular resolution.
His construction is based on a Voronoi diagram for circle packings that is
invariant under Möbius transformations (which are known to be anglepreserving; see e.g. Bern and Eppstein [127]). The weighting scheme used is
∆ (x, p) =
d(x, p)2 − r(p)2
,
2r(p)
where p and r(p) are the center and radius, respectively, of a circle in
the packing. Figure 7.22 gives a small example of a Lombardi drawing.
Interestingly, the same construction can be used to investigate the
graph-theoretic properties of soap bubbles. In the arising spherical cell
complexes, angles around edges and vertices are equally spaced as well,
by Plateau’s laws which have been proved by Taylor [680] in 1976.
From their piecewise spherical shape, soap bubbles actually look like the
regions of a 3D multiplicatively weighted Voronoi diagram, the weights
reflecting pressure conditions; but clearly, no disconnected regions can
occur. Whether this naive conjecture is justified is still unsettled, though.
The graphs formed by two-dimensional ‘soap bubbles’ have been recently
characterized in Eppstein [325], based on his previously mentioned results
in [324].
Here may be the right place to observe that the distance cone (lower
envelope) model for Voronoi diagrams is not identical to the growth model
(i.e., the wavefront model ), where the expansion of circles, or of any
other appropriate shapes, ceases when getting in contact with each other.
The latter model cannot treat Voronoi diagrams with disconnected regions.
Figure 7.22. Lombardi drawing of a graph with seven vertices. It contains vertices of
, and π2 , respectively.
degree 2, 3, and 4, with circular edges spaced at angles π, 2π
3
162
Chapter 7. General Spaces & Distances
A discussion of this issue, in the context of straight skeletons, is given
in [38].
In fact, when modeling certain phenomena in the natural sciences,
letting cease the expansion of circles once they start overlapping is more
appropriate. For example, crystals growing from sites at different speeds
will still produce connected regions. The same applies to colonial growth
patterns of plants.
Schaudt and Drysdale [617] first studied this interesting variant and
called it the crystal growth Voronoi diagram. Regions in this diagram
may wrap around other sites’ regions — hence they may not be simply
connected — and shortest path lengths from sites to points are taken to
measure distance then, considering completed regions as obstacles — hence
regions will always be connected. This ‘dynamic mixture’ of multiplicatively
weighted Voronoi diagram and geodesic Voronoi diagram (Section 5.4) leads
to a quite complicated structure. Its worst-case size is Θ(n2 ), even though
there are only n regions, because the complexity of the site separators can
be high, combinatorially as well as geometrically. (For instance, a quadratic
size occurs if sites of increasing weights are situated on a straight line.)
An exact O(n3 ) time construction algorithm is given in the paper above;
however, separators have to be determined by numerical methods, as no
closed form is available.
Kobayashi and Sugihara [471, 472] present a maybe more practical
digital algorithm, which approximates the crystal growth Voronoi diagram
in pixel representation (cf. Section 11.1). They also mention another
application, concerning collision-free path planning for a robot avoiding
enemy attacks.
We note that for additive weights, i.e., same growth rate for all circles
but possibly different starting times, there is no difference between the
growth model and the distance cone model: Regions are star-shaped in the
additively weighted Voronoi diagram, and geodesic paths from a site p to
any of the points x in its region are just straight-line segments, because x is
always visible from p. Therefore, it is not meaningful to define a ‘crystal
growth version’ of additively weighted Voronoi diagrams; no change in the
structure will happen.
Multiplicatively weighted Voronoi diagrams with visibility constraints
have been considered in Wang and Tsin [698], in the presence of line segment
obstacles. As the unweighted variant of this diagram is fairly complex
already, see Section 5.4, weighting has (almost) no effect on its size and
construction time.
7.4. Weighted distance functions
163
7.4.4. Anisotropic Voronoi diagrams
Here we consider an interesting and recently introduced diagram, where
each site p is weighted by an individually assigned distance function,
instead of a real number w(p). The resulting anisotropic Voronoi diagram
is investigated in Labelle and Shewchuk [479]. Each site p is allotted with
a positive definite quadratic form, given by a square matrix Mp = FpT Fp ,
where the positive definite matrix Fp maps the physical plane into a rectified
plane with the usual length, area, and angle measures, as seen by site p. In
particular, the distance between two points x and y in the plane (from p’s
view) is
dp (x, y) = (x − y)T Mp (x − y).
The Voronoi region of p is defined to contain all points x in the plane
where dp (x, p) ≤ dq (x, q), for all sites q different from p. Anisotropic
Voronoi diagrams are a generalization of multiplicatively weighted Voronoi
diagrams, where the matrices Mp are just multiples of the unit matrix I,
Mp = w(p) · I
for multiplicative site weights w(p). They exhibit an according behavior,
with their regions being possibly disconnected and multiply adjacent.
Bisectors are general quadratic curves, i.e., all types of conics can occur, as
opposed to (Apollonius) circles. See Figure 7.23 for an illustration.
The diagram’s combinatorial complexity is bounded by O(n2+ε ), by an
interpretation in the lower envelope model in R3 . Surprisingly, a totally
different embedding in 5 dimensions exists; see Boissonnat et al. [150].
The anisotropic Voronoi diagram can be obtained by intersecting a certain
manifold and a power diagram in R5 .
In the main fields of application, namely, interpolation of functions and
numerical modeling [479], an appropriate dual of the diagram is sought, in
order to allow to minimize the number of triangles in the anisotropic mesh
while retaining a good accuracy in computations. However, the geometric
dual of an anisotropic Voronoi diagram is not a triangulation of the given
sites, in general, due to the presence of disconnected and/or multiply
adjacent regions. Conditions under which the dual is a correct triangulation
(the so-called anisotropic Delaunay triangulation) are given in [479], along
with respective construction algorithms that insert extraneous sites at wellchosen locations.
An alternative site insertion algorithm for computing anisotropic
Delaunay triangulations, based on power diagrams, is proposed in [150].
164
Chapter 7. General Spaces & Distances
Figure 7.23. Anisotropic Voronoi diagram for 10 sites. Regions are patterned by isocontours of the distance function for the closest site. While every site is contained in
its region, not every face of the diagram contains a site, due to region disconnectedness
(from [479]).
See also Canas and Gortler [178], where further sampling and distribution
conditions on the sites are given, in order to make the regions in an
anisotropic Voronoi diagram connected (i.e., orphan-free). In another
paper [179], they show that orphan-free anisotropic Voronoi diagrams have
duals that are indeed triangulations.
Naturally, anisotropy in the plane can be specified in various ways,
leading to different types of generalized Voronoi diagrams. Du and
Wang [290] use an anisotropic metric field to define, for each point x ∈ R2 ,
an individual anisotropic distance function. In contrast to [479], Voronoi
regions are computed by using the complete metric. Kunze et al. [478]
generate regions by means of geodesic Voronoi diagrams (Subsection 7.1.1)
on parametric surfaces. A surface embedding of the plane in higherdimensional space is also used in Kapl et al. [441], who define the distance of
two points in R2 as the Euclidean distance in Rd of the two corresponding
points on the respective surface. This has the advantage that regions can be
obtained by intersecting the surface with a classical Voronoi diagram in Rd .
The embedding is generated, via a fitting procedure, from a given distance
graph in R2 that specifies the anisotropy. One more way of introducing
anisotropy is by weighting differently certain subsets of the plane; see
Subsection 7.6.1.
7.4. Weighted distance functions
165
7.4.5. Quadratic-form distances
Let us say at this place a few more words about general quadratic-form
distances in Rd , whose positive instances have been used to define the
anisotropic Voronoi diagram in R2 . Such distances are intimately related
to the weighted concept of power diagrams. They are of the form
Q(x, y) = (x − y)T · M · (x − y),
where M is a fixed non-singular d × d matrix. We may assume that, without
loss of generality, M is symmetric, as we have Q(x, y) = 12 (x − y) · (M +
M T ) · (x − y). Interesting classes of Voronoi diagrams are induced, with
surprising applications.
Separators are hyperplanes in Rd . Therefore, such diagrams are
polyhedral cell complexes, which are sometimes called affine Voronoi
diagrams in the literature. We observe that the choices M = I and M =
−I produce the classical Voronoi diagram and its farthest-site variant,
respectively.
Unified treatments are given in Edelsbrunner and Seidel [310] via
Voronoi surfaces in Rd+1 (Subsection 7.5.1), and in Aurenhammer and
Imai [98] via power diagrams in Rd (Subsection 6.2.3), asserting that the
class of diagrams obtainable by Q coincides with the class of affine Voronoi
diagrams and also with the class of power diagrams.
For d = 2 and M = (01 10 ), the distance Q(x, y) describes the area of the
isothetic rectangle with diagonal xy. The obtained diagram proves useful
in finding the largest empty axis-aligned rectangle among a finite set of
points in the plane; see Chazelle et al. [196]. More recently, Aurenhammer
et al. [104] gave attention to the quasi-Euclidean distance generated by the
special Jordan matrix
M = diag(1, . . . , 1, −1).
This distance is capable of embedding the Voronoi diagram for oriented
spheres in Rd−1 into a power diagram in Rd . Namely, if the dth coordinates
of two points p, q ∈ Rd are interpreted as ‘radii’ (which are possibly
negative), and the first d − 1 coordinates of p and q specify centers, then
Q(p, q) expresses the squared tangent length between the respective two
oriented spheres in Rd−1 .
The aforementioned rectangle area related distance transforms to the
two-dimensional instance of Q with M = diag (1, −1) by a rotation of π4 .
Motivation for studying quasi-Euclidean distances comes from special
relativity theory, see e.g. Benz [123] and Woodhouse [707], where one
166
Chapter 7. General Spaces & Distances
describes events as points in the quasi-Euclidean space whose ‘metric’ is
governed by the matrix M , and whose isometric mappings are the Lorentz
transformations. The line spanned by two events p and q can be time-like, or
space-like, or light-like — these cases correspond to Q(p, q) < 0, Q(p, q) > 0,
and Q(p, q) = 0, respectively. The value of Q(p, q) is a Lorentz invariant:
If positive, it has an interpretation as square of distance, and if negative,
then the square root of its absolute value is the proper time experienced by
a particle whose life line is the line segment pq.
Several physically meaningful modifications are considered in [104]. For
example, the absolute-valued variant ∆A (p, q) = |Q(p, q)| models a setting
where lifetime is regarded as a positive distance. Or, putting
∆∞ (p, q) =
Q(p, q) if ≥ 0,
∞
otherwise
lets oriented spheres to which squared tangent lengths are negative not take
part in the local formation of Voronoi regions — a ‘space-driven’ variant.
When using
∆0 (p, q) =
Q(p, q) if > 0,
0
otherwise,
negative squared tangent length means being as close as possible. That
is, lifetime of particles is irrelevant again, apart from the fact that it
restricts the diagram to domains where distances to all oriented spheres are
positive.
Figure 7.24 depicts a partition of the plane induced by the distance
variant ∆A and three oriented spheres in R1 , interpreted as point
sites in R2 . Two types of separators arise, namely, for the equation
Q(x, p) = Q(x, q), which gives straight lines, as well as for the equation
Q(x, p) = −Q(x, q), which gives hyperbolas. Colors gray, light gray, and
white symbolize ownership of region parts by individual sites. The diagram
has a size of Θ(n2 ) and it can be constructed within this time bound,
by a close structural connection to the underlying orthogonal straight line
grid G, that reflects the sign pattern of Q for the given n sites (drawn in
dotted style).
The same grid G also structures the diagram for the distance ∆∞ ,
illustrated in Figure 7.25. In fact, the diagram is a linear refinement of G;
each of its grid cells Z is partitioned by the power diagram of exactly those
sites whose distances to Z are positive. (The same color code for ownership
applies; the topmost and the bottom-most cell, shown hatched, belong to
7.5. Abstract Voronoi diagrams
167
111111111111
000000000000
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
000000000000
111111111111
2
1
3
1111111111111
0000000000000
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
0000000000000
1111111111111
Figure 7.24.
Distance ∆A .
Figure 7.25.
Distance ∆∞ .
no site.) A combinatorial upper bound of O(n3 ) results which, however, we
do not consider as tight.
For applications in physics, diagrams in space-time R4 are needed. The
sign grid for Q now is an arrangement of homothetic rotational doublecones, rather than of linear double-wedges, which complicates matters. Its
size is Θ(n4 ), and that of a power diagram is Θ(n2 ) in the worst case
(by Theorem 6.1), piling up to a — probably not tight — upper complexity
bound of O(n6 ). Of somewhat simpler nature is the diagram induced by the
distance ∆0 . It basically consists of two power diagrams in R4 , each clipped
by a cone. A combinatorial complexity and construction time of Θ(n2 ) can
be shown.
7.5. Abstract Voronoi diagrams
One might wonder if there exist some intrinsic properties most Voronoi
diagrams have in common, and if these properties are a foundation strong
enough for a unifying approach to the definition and to the construction of
Voronoi diagrams.
7.5.1. Voronoi surfaces
Edelsbrunner and Seidel [310] have introduced a very general approach.
It does not explicitly use the concepts of sites and distance functions.
Rather, the site set, S, is a mere set of indices {p, q, . . .}. A fixed domain, X,
is given, and for each p ∈ S a real-valued function fp : X → R. The graph
Hp = {(x, fp (x)) | x ∈ X}
168
Chapter 7. General Spaces & Distances
of fp can be considered a hypersurface in the space X × R, sometimes
referred to as the Voronoi surface of site p. Now the Voronoi diagram of
(S, {fp | p ∈ S}) is defined to be the projection onto X of the lower
envelope
L=
x, min fp (x) x ∈ X
p∈S
of the arrangement {Hp | p ∈ S} of hypersurfaces. This diagram is also
called the minimization diagram of the functions fp , as L is their pointwise
minimum.
At the end of Section 3.5, and quite extensively in the preceding
Section 7.4, we have already studied several examples of this approach.
Section 6.2 shows that there exist even linear functions fp whose envelope
projects onto the Voronoi (or power) diagram of S. Moreover, the order-k
Voronoi (or power) regions of S are the projections of certain cells of the
corresponding arrangement; see Section 6.5.
To establish this connection between Voronoi diagrams and arrangements of hypersurfaces in a space one dimension higher is one of
the great advantages of this approach. As one implication, an upper
bound of O(nd+ε ) on the combinatorial complexity of general classes of
Voronoi diagrams follows from a result in Sharir [639] on envelopes of
hypersurfaces; see Sections 6.6 and 7.2. In general, the computation of
lower envelopes, and the analysis of their combinatorial complexity, is a
central problem in computational geometry; see e.g. the book by Sharir and
Agarwal [642].
7.5.2. Admissible bisector systems
A different approach to planar Voronoi diagrams has been suggested in
Klein [461] and improved on in Klein et al. [462]. Again, there are no
physical sites but a finite index set, S. Instead of distance functions or
their graphs in 3-space, bisecting curves (separators) are used as primary
objects.
Let us assume that for any two different indices p, q in S, a bisecting
curve J(p, q) = J(q, p) is given, which cuts the plane into two open domains,
D(p, q) and D(q, p). We assume that these curves are nice, in the sense of
Definition 7.1. This implies in particular that each curve J(p, q) and the
domains D(p, q), D(q, p) are unbounded. In the subsequent figures, the
index p denotes the ‘halfplane’ D(p, q).
The Voronoi region VR(p, S) is defined as the intersection of the open
domains D(p, q), where q ∈ S \ {p}. The abstract Voronoi diagram (AVD
7.5. Abstract Voronoi diagrams
169
for short) V (S) of {J(p, q) | p, q ∈ S, p = q} is defined as the edge graph
constituting the complement of all Voronoi regions.
Definition 7.7. The bisector system J = {J(p, q) | p, q ∈ S, p = q}) is
called admissible iff, for each subset S ⊂ S of size at least 3, the following
conditions are fulfilled.
(1) The Voronoi regions VR(p , S ), where p ∈ S , are path-connected.
(2) Each point of the plane lies in the closure of a Voronoi region VR(p , S ).
Figure 7.26(i) shows an admissible curve system for S = {p, q, r},
and the resulting abstract Voronoi diagram (ii). We observe that the
requirements of Definition 7.7 allow for some phenomena that cannot occur
for Euclidean bisectors of points. For example, point a lies in the intersection
of two bisecting curves J(p, r) and J(q, r), but not on the third curve,
J(p, q). Point w does lie on all three bisecting curves, but it is not a
Voronoi vertex. The intersection of two bisecting curves, J(p, r) and J(q, r),
p
q
ai ai+1 a
r
p
r
q
r
q
p
q
r q
p
q
v
r
p
r r
p q
r
q
w
p
r
q
p
(i)
VR(r,S)
v
VR(p,S)
VR(q,S)
(ii)
Figure 7.26.
diagram (ii).
An admissible curve system (i) and the resulting abstract Voronoi
170
Chapter 7. General Spaces & Distances
consists of an infinite number of isolated points that converge towards
point a.
Still, the conditions stated in Definition 7.7 are strong enough to obtain
a satisfying structure theory and efficient algorithms for AVDs. Some wellknown structural properties of ‘concrete’ Voronoi diagrams can be derived
directly. For example, the Voronoi diagram consists of all points in the plane
that belong to the closures of two or more Voronoi regions; cf. Definition 2.1.
With concrete distance functions, if a point z is closer to site p than
to q, and closer to q than to r, then, trivially, z is closer to p than
to r. The following lemma states an analogous fact for abstract Voronoi
diagrams.
Lemma 7.7. Let J be an admissible system, and let p, q, r ∈ S. Then,
D(p, q) ∩ D(q, r) ⊆ D(p, r) holds.
Proof. Let z ∈ D(p, q) ∩ D(q, r). Point z must be contained in one of the
sets D(r, p), J(r, p), D(p, r). If z were contained in D(r, p), it could not lie
in any of the closed Voronoi regions
VR(q, S ) ⊆ D(q, p) = D(q, p) ∪ J(q, p),
VR(r, S ) ⊆ D(r, q) = D(r, q) ∪ J(r, q),
VR(p, S ) ⊆ D(p, r) = D(p, r) ∪ J(p, r)
for S = {p, q, r}. This is impossible since these three sets cover R2 , by the
third requirement in Definition 7.7.
Suppose z ∈ J(r, p). By the Jordan property of J(r, p), there exists an
arc α with endpoint z such that α\{z} is fully contained in D(r, p). With z,
even a neighborhood U of z is contained in the open set D(p, q) ∩ D(q, r).
Inside U , path α contains a point z ∈ D(p, q) ∩ D(q, r) ∩ D(r, p), which
leads to the same contradiction as before. Consequently, the third case
applies, that is, z ∈ D(p, r).
Lemma 7.7 is useful in proving that an abstract Voronoi diagram V (S)
is a planar graph with O(|S|) many faces. Its vertices are just the points
contained in the closures of three or more Voronoi regions. For algorithmic
purposes, one can turn V (S) into a bounded structure, by encircling it
with a large closed curve, as proposed in Section 2. In this way it becomes
a 2-connected planar graph with at most n + 1 faces, all of whose vertices
have degree at least 3. Each such graph can be obtained as an abstract
Voronoi diagram.
Applying the concept of abstract Voronoi diagrams to a concrete
situation is facilitated by the following result.
7.5. Abstract Voronoi diagrams
171
Theorem 7.10. A curve system J is admissible if the properties stated
in Definition 7.7 are fulfilled for each subset of 3 sites of S.
The conditions in Definition 7.7 are met, for example, by the bisectors of
power diagrams (Section 6.2), of sites with additive weights (Section 7.4),
and of non-intersecting line segments or convex polygons (Section 5.1);
see Meiser [539]. Another example has been provided by Edelsbrunner
et al. [304] and Abellanas et al. [5]. They studied disjoint compact and
convex sites in the plane, and their Hausdorff Voronoi diagram based on a
convex distance function dC , which puts each point x into the Voronoi
region of the first site that is fully contained in a shape C expanding
from x; cf. Section 7.2. It can be shown that the resulting regions are simply
connected.
Also, the types of Voronoi diagrams studied by McAllister et al. [527],
Ahn et al. [22], Karavelas and Yvinec [446], Abellanas et al. [6], Aichholzer
et al. [38], and Bae and Chwa [109] are special cases of abstract Voronoi
diagrams.
As a general result, we have the following lemma for the class of all very
nice metrics (in the sense of Definition 7.4). We include the proof which
has not yet appeared elsewhere.
Lemma 7.8. If m is a very nice metric then its bisecting curves defined
by Definition 7.5 form an admissible system.
Proof. To prove that the resulting Voronoi regions are pathwise connected,
let z ∈ VR(p, S). Since VR(p, S) is open, there is a closed disk D such that
z ∈ D ⊂ VR(p, S) holds. Due to
|m(p, v) − m(p, w)| ≤ m(v, w)
and because of Definition 7.2(1), the function g(w) = m(p, w) is continuous.
Thus, g attains its maximum value in the compact set D at some point
c ∈ D. Let πc denote an m-straight path from p to c. As homeomorphic
image of a line segment, πc cannot be space-filling: there must be a point
b ∈ D that does not lie on πc . Conversely, c itself cannot be situated on
an m-straight path from p to b, by maximality. Therefore, Definition 7.4(3)
applies, that is, there are m-straight paths ρb and ρc from p to b and c,
respectively, that have only point p in common. They are contained in the
closure of VR(p, S), by Lemma 7.4.
If we clip ρb and ρc at their first points situated on the boundary ∂D
of D, they form, together with a segment of ∂D, a Jordan curve whose
interior, I, is contained in VR(p, S), and we can easily route a path from z
to p through I.
172
Chapter 7. General Spaces & Distances
The proof of Property (2) of Definition 7.7 is by straightforward case
analysis, using Theorem 7.10 and Lemma 13 in [462].
Thus, the AVD machinery applies to all very nice metrics in the plane.
If a metric is nice, but not very nice, Lemma 7.8 may not hold; for
example, the curves J(p, q) and J(q, r) shown in Figure 7.16 cannot form an
admissible system because D(q, p) ∩ D(q, r) would be disconnected. In this
case, a more involved version [461] of AVDs could be applied, or one could
resort to perturbation techniques.
There are other concrete Voronoi diagrams which are not covered by
the AVD concept, because their bisecting curves fail to fulfill some of the
properties stated in Definition 7.7 above. For example, in the closest-polygon
Voronoi diagram, bisectors may be closed loops while its regions are still
path-connected. In the farthest-segment Voronoi diagram investigated by
Aurenhammer et al. [95], bisectors are unbounded simple curves, as required
in Definition 7.7, but Voronoi regions may be disconnected. In a more
general setting, namely for the farthest-polygon Voronoi diagram explored
by Cheong et al. [208], even closed loop bisectors and disconnected regions
may occur, while the diagram is still of linear complexity. In the Voronoi
diagram of points with multiplicative weights both properties are violated
as well, and its complexity may be quadratic; see Section 7.4.
7.5.3. Algorithms and extensions
Of the four algorithmic methods mentioned in Chapter 3 for computing
the Euclidean Voronoi diagram, two have been adapted to abstract Voronoi
diagrams.
How to generalize the divide & conquer approach has been shown
in [461]. As in the case of nice metrics (Section 7.3) one has to assume that
the index set, S, can recursively be split into subsets L and R such that the
merge chain B(L, R) contains no cycle, in order to find the starting edges
of B(L, R) at infinity. Once a starting edge has been found, B(L, R) can be
traced efficiently, because abstract Voronoi regions are pathwise connected;
no stronger properties, like star-shapedness or convexity, are needed. This
is illustrated in Figure 7.27, which should be compared to Figure 3.7.
It has been shown in Mehlhorn et al. [533] and in Klein et al. [466] that
one can, without further assumptions, apply the randomized incremental
construction technique to abstract Voronoi diagrams.
Theorem 7.11. The abstract Voronoi diagram of a given admissible
system J = {J(p, q) | p, q ∈ S and p = q}, where |S| = n, can be
7.5. Abstract Voronoi diagrams
173
vL,2
p
VR(p,L)
p
q2
p
p1
q2
vR
p
q1
vL
p
v
q1
Figure 7.27. If J(p, q2 ) ran like this, the disconnected shaded areas would both belong
to VR(p, {p, p1 , q2 }), a contradiction. Therefore, the first point, vL,2 , on the boundary
of VR(p, L) can still be found by scanning the boundary from vL on counterclockwise.
constructed within expected O(n log n) many steps and expected space O(n),
by randomized incremental construction.
This algorithm provides a universal tool that can be used for computing
all types of concrete Voronoi diagrams whose bisector curves have the
properties required in Definition 7.7. To adapt the algorithm to a special
type, one has to implement only certain elementary operations on bisector
curves. For example, it is sufficient to have a subroutine that accepts
five elements of S as input, and returns the graph structure of their
abstract Voronoi diagram. All numerical computations can be kept to this
subroutine.
It seems difficult to apply the plane sweep approach to abstract Voronoi
diagrams. One reason is the absence of sites whose detection by the sweep
line could indicate that a new region must be started.
Abstract Voronoi diagrams can also be thought of as lower envelopes,
in the sense mentioned in Subsection 7.5.1. Namely, for each point x not
situated on a bisecting curve, the relation
p <x q
iff x ∈ D(p, q)
defines a total ordering on S, thanks to Lemma 7.7. If we construct a set
of surfaces Hp , p ∈ S, in 3-space such that Hp is below Hq iff p <x q holds,
then the projection of their lower envelope equals the abstract Voronoi
diagram.
Chapter 7. General Spaces & Distances
174
r
q
p
p
q
r
p
q
q
p
Figure 7.28. Abstract inverse Voronoi diagram for the bisectors of three sites p, q, and r.
The region of p is not connected.
The concept of abstract Voronoi diagrams has been generalized
to 3-space in Lê [488], under structural restrictions. For example,
any two bisecting surfaces J(p, q), J(p, r) must have an intersection that
is homeomorphic to a line, or empty. These assumptions apply, e.g., to
convex distance functions whose unit sphere is an ellipsoid ; cf. Section 7.2.
Rasch [602] and Mehlhorn et al. [534] have studied abstract inverse
(or farthest-site) Voronoi diagrams in the plane; cf. Section 6.5. Here the
region of p is defined as the intersection of all open domains D(q, p),
where q ∈ S \ {p}. In contradistinction to the Euclidean case, the resulting
regions are in general not pathwise connected; see Figure 7.28.
If the admissible system J is such that none of the regions in the
(closest-site) abstract Voronoi diagram generated by J is empty, then the
inverse diagram with respect to J is connected. It is a tree of complexity
O(n) and can be constructed within expected O(n log n) many steps and
expected linear storage.
Bohler et al. [137] have recently introduced abstract Voronoi diagrams
of arbitrary order k. For a subset M of size k of S, its abstract order-k
region is defined as the intersection of all domains D(p, q), where p ∈ M
and q ∈ S \ M ; cf. Section 6.5. Such order-k regions need not be connected;
even for k = 2 a region can consist of up to n − 1 faces.
Let us assume that the bisector system J is in general position, so that
no more than three curves pass through one point and all intersections are
transversal. Moreover, suppose that the standard order-1 abstract Voronoi
regions are non-empty. Then the following upper bound can be shown [137].
Theorem 7.12. Under the conditions above, an abstract order-k Voronoi
diagram has at most 2k(n − k) many faces, and this bound is tight.
This result generalizes and sharpens the O(k(n − k)) upper complexity
bounds for the order-k diagrams of points and line segments; see Lee [495]
7.6. Time distances
175
and Papadopoulou and Zavershynskyi [584], respectively. Not covered is
the case of point sites under the geodesic distance in a polygon with holes,
which has recently been studied by Liu and Lee [510].
Testing order-1 abstract Voronoi regions for non-emptiness is facilitated
by the following lemma complementing Theorem 7.10. If all bisecting curves
J(p, q) were lines, the claim would follow directly from Helly’s theorem
(see e.g. [371]) on intersections of convex sets.
Lemma 7.9. In order to verify that all Voronoi regions VR(p, S ), where
S ⊆ S, are non-empty it is sufficient to consider the subsets S of size 4.
7.6. Time distances
7.6.1. Weighted region problems
Imagine a large modern-style city, with streets arranged in north–south
and east–west directions. The city is equipped with a public transportation
network like a subway or a bus system. Time is precious and people intend
to follow the quickest path from their homes to their desired destinations,
using the network whenever appropriate. For some people several facilities
of the same kind are equally attractive (think of post offices or hospitals),
and their wish is to find out which facility is reachable first. There also may
be commercial interest (from real estate agents, or from a tourist office) to
make visible the area which can be reached in, say one hour, from a given
location in the city (the apartment for sale, or the recommended hotel).
Applications like this lead to Voronoi diagram models where distances
are measured differently at certain places of the plane, rather than for
different sites, as it is the case in the weighted model (in Section 7.4):
Within the transportation network, movement is quicker, so distances are
shorter.
In its general form, such a model is treated in Mitchell and
Papadimitriou [548]. They consider a partition of the plane into weighted
regions, the weight expressing internal travelling speed (or its reciprocal, the
time per unit distance). As a basic task, for two given points p and q the
quickest (Euclidean) path between p and q across the weighted regions is
sought, entering and exiting regions at any place. The solution in [548] is
based on Snell’s law of refraction. The query time is O(t7 · ), where t is
the complexity of the polygonal partition and denotes the precision of the
problem instance.
Allowing only weights 0 and 1 leads to the shortest (obstacle-avoiding)
path problem (or geodesic distance problem), for which much more efficient
176
Chapter 7. General Spaces & Distances
solutions are known; see e.g. Mitchell [546], and Section 5.4 for the resulting
geodesic Voronoi diagram.
A given transportation network T in the plane can be viewed as a
one-dimensional instance of the weighted region problem. Choose a weight
ve > 1 for each network segment e, and weight 1 on all connected parts
of the complement of T in the plane. This variant (and others) have been
studied by Gewali et al. [360], who construct the quickest path between two
given points p and q in time O(t2 ), where t denotes the complexity (number
of edges) of the network.
The air-lift distance concept in Section 7.3 constitutes a zerodimensional variant; distances are shorter only between points in a given
finite set.
7.6.2. City Voronoi diagram
In the following, we will present the special one-dimensional case where
the transportation network T is isothetic and of constant speed v, and
where the underlying metric is the L1 -norm. This exactly models the
practical situation mentioned at the beginning of this section, and has been
investigated in Aichholzer et al. [38].
For two points x and y in the plane, let QT (x, y) denote a quickest path
between them, that is, a path minimizing the travel duration between x
and y, using T . Let dT (x, y) be the time for traversing QT (x, y). Note first
that dT induces a metric in the plane, which we shall call the city metric:
dT is non-negative, symmetric, and obeys the triangle inequality because,
for each point z ∈ QT (x, y), the concatenation of QT (x, z) and QT (z, y)
gives QT (x, y). Quickest paths are isothetic polygonal lines which, apart
from special cases, are not unique. A single path may contain Θ(t) pieces
on T plus Θ(t) shortcuts (foot walks outside T ). The complexity of the city
metric is also apparent from the shape of the induced circles
C(x, r) = {y | dT (x, y) = r}.
Not only may C(x, r) break into Θ(t) pieces, its shape also varies with its
radius r, as well as with the relative position of its center x to T . Still, the
‘disk’ interior to C(x, r) is a connected set, as being the union of quickest
paths.
Let now S be a set of n point sites in the plane. The city Voronoi
diagram VT (S) contains, for each site s ∈ S, the region of all points given by
reg(s) = {x | dT (x, s) < dT (x, u), ∀u ∈ S\{s}}.
7.6. Time distances
177
All regions are path-connected (and in fact, the diagram consists of simply
connected regions), by the fact below.
Observation 7.1. reg(s) is the union of all quickest paths from x ∈ reg(s)
to the site s, and equivalently, to the set S.
Observation 7.1 does not imply that VT (S) is of size O(n). Vertices
of degree 2 may occur, whose number cannot be bounded by applying the
Eulerian theorem to a planar graph with n faces. A trivial upper bound is
O(n · t), where t counts the number of edges of the network T . The true
complexity of VT (S) is only O(n + t), as we shall see later.
Let us consider the bisector of two sites s, u ∈ S in the city metric,
which is the set
Bsu = {x | dT (x, s) = dT (x, u)}.
The set Bsu is connected by Observation 7.1. However, due to wellknown L1 -specific degeneracies (see Section 7.2), Bsu may fail to be
one-dimensional. A canonical one-dimensional form of Bsu results from
the straight skeleton interpretation of VT (S) given next. In this form,
Bsu is a (possibly cyclic) polygonal curve with Θ(t) edges in the worst
case. Unfortunately, cyclic and non-constant sized bisectors are known to
be a main obstacle for efficiently applying the construction techniques
divide & conquer (Section 3.3) and randomized incremental insertion
(Section 3.2).
There is a useful link between the city Voronoi diagram and straight
skeletons introduced in Section 5.3. Its implications are manyfold [38].
To aid the intuition, let us recall the interpretation of the standard
L1 -metric Voronoi diagram in the growth model. Imagine each site in the
given set S sends out a wavefront, in the shape of an expanding√L1 circle
(called a diamond ), at time 0 and with translational speed 1/ 2 for all
its edges. Wavefront movement stops at every point x where two diamonds
come into contact, and x is declared a point on the edge graph of the
L1 -Voronoi diagram.
This expansion view actually shows that every L1 -Voronoi diagram for
point sites is a straight skeleton, with these diamonds as its defining figures.
How does the network T influence this growth model? The circles
expanding around the sites change their shapes, in a manner determined by
T which we want to explore next. Let s ∈ S be a fixed site, and recall that
C(s, r) denotes the circle of radius r around s in the city metric. Our aim
is to interpret C(s, r) as the common wavefront propagated from a certain
set Fs of figures.
178
Chapter 7. General Spaces & Distances
Figure 7.29. Co-centric dT -circles around a site s. The small black polygons depict the
generated figures in the set Fs (from [38]).
See Figure 7.29. For sufficiently small r > 0, C(s, r) is a diamond with
center s as before. Let Fs be a set of figures which initially contains this
diamond, associated with its place s and time 0 of birth. Now consider all
points x ∈ T where either a vertex of C(s, r) runs into a network segment,
or an edge of C(s, r) slides into a network node. At time r = dT (s, x), and
at no other point in time, the circle C(s, r) changes its combinatorial shape.
Accordingly, we add to the set Fs an appropriate figure with time r and
place x of birth, constructed as follows.
Observe that, at time r + ε, a diamond D with center x and diagonal
length ε appears on C(s, r), along with sharp-angled wedges tangent to D,
whose peaks move at speed v in all possible directions on T that point to
the exterior of C(s, r). The figure we construct is simply the union of D and
these wedges. Each wedge is an isosceles triangle with base b and height
ε · v, where b is a diagonal of D.
We categorize the figures in Fs by their number j of wedges and call
them j-needles. Note that 0 ≤ j ≤ 3 holds, because the degree of a network
node is at most four, and at least one of the four isothetic directions points
to the interior of C(s, r); see Figure 7.29 again.
Let Wr (Fs ) denote the common wavefront of all the figures in Fs .
By construction, Wr (Fs ) enjoys the following property: Wr (Fs ) coincides
with C(s, r) at any time r. This property has an obvious but important
consequence: It draws the connection to straight skeletons.
7.6. Time distances
179
Theorem 7.13. Let F = s∈S Fs . Then the straight skeleton SK(F ) of
F contains the city Voronoi diagram VT (S) as a subgraph.
The set F in Theorem 7.13 contains redundant figures, i.e., figures which
do not contribute to SK(F ). Their wavefronts solely and always consist of
points swept over earlier (or at the same time) by wavefronts of other figures.
For example, for each site s ∈ S, all figures of Fs whose places of birth lie
outside the region reg(s) in VC (S) are redundant in F . Edges of SK(F )
which are not edges of VC (S) stem from figures born in the same region of
VC (S). We will alternatively call SK(F ) the refined city Voronoi diagram
of S and C, and denote it with VC (S). The two-dimensional components
of VC (S) are its regions, which are the regions of VC (S), and which are
partitioned into faces, the faces of SK(F ); see Figure 7.30.
The diagram VT (S) = SK(F ) is a planar graph whose faces are
connected and whose vertices have degree at least three, as in every straight
skeleton. The number of edges and vertices of VT (S) therefore is linear in
the number of its faces. This number, in turn, is bounded by the total
number of edges of all figures which are non-redundant in the set F , because
each such edge (possibly) sweeps out a single face of VT (S).
How many figures are non-redundant? Clearly, each site s ∈ S defines
a non-redundant diamond with birth place s. Moreover, each node of C
gives rise to a single non-redundant j-needle born at this node. This gives
n + t relevant figures so far. It remains to examine the 2-needles born in
the interior of network segments. Their number is Θ(n · t) in the worst case:
each of the n sites may cause t such figures, born on t parallel segments
of T ; However, if f is some interior 2-needle on network segment σ, then
the interior 2-needle f born when the wavefront of f hits some segment
parallel to σ is redundant in the set {f, f }.
Redundant needles of the last kind indicate unused portions of T
(dotted in Figure 7.30). They should not appear in an efficiently designed
network.
An interior 2-needle never stems from a contact between a network
segment and some sharp-angled vertex of a needle. Therefore, only the
interior 2-needles that stem from contacts with vertices incident to some
diamond edge are non-redundant in the set F . As such vertices disappear
from the common wavefront of F after their first contact with T , we have at
most 4(n + t) figures of this kind. In summary, F contains at most 5(n + t)
non-redundant figures. Each figure is of constant complexity, which gives a
total of O(n + t) figure edges, and with it, skeleton faces.
180
Chapter 7. General Spaces & Distances
Figure 7.30. The city Voronoi diagram (black thin lines) for five sites and an isothetic
transportation network (bold lines). The faces of the diagram are hatched by iso-contours
for the city metric (gray lines); from [38].
Theorem 7.14. The refined city Voronoi diagram VT (S) consists of
O(n + t) faces, edges, and vertices.
7.6.3. Algorithm and variants
On the algorithmic side, all non-redundant figures can be determined in
time O(n log n + t2 log t), by exploring the isothetic grid defined by the
vertices of T with the continuous Dijkstra technique. It remains to construct
the straight skeleton of these figures in an efficient way.
7.6. Time distances
181
Fortunately, the abstract Voronoi diagram machinery (Section 7.5)
can be employed in this special case — and unlike for general straight
skeletons — in spite of the fact that the bisector system defined by taking
these figures as abstract sites is not admissible: Areas belonging to no
abstract Voronoi region (no-man’s lands) may arise, and bisectors may
be closed curves. These shortcomings can be remedied, by modifying the
figures so as to behave accordingly while leaving their straight skeleton
unchanged; see [38] for details.
An abstract Voronoi diagram with m sites and constant bisector
complexity (as it is the case for our figures) can be computed
in time O(m log m). Particularly attractive is a randomized incremental
construction; see Mehlhorn et al. [533] and Klein et al. [466]. To adapt this
algorithm to our situation, it suffices to have a subroutine that accepts five
figures as input and returns their straight skeleton. Any trivial skeleton
algorithm may be implemented in this subroutine, because all input figures
are convex, 12-oriented, and have at most six edges.
In summary, since m = O(n + t) holds by Theorem 7.14, the time for
finding the figures dominates, and O(n log n + t2 log t) time (and optimal
O(n + t) space) is taken for constructing the city Voronoi diagram.
In the paper [374], Görke et al. showed how to find the figures more
efficiently, in O((n + t) polylog (n + t)) time, which is faster for large t.
Naturally, time distances for the Euclidean metric (rather than the
L1 -metric) are of interest in several applications. For the basic case
where the transportation network T is a single straight line (a highway,
for example), the resulting Voronoi diagram was studied in Abellanas
et al. [7]. Lee et al. [498] generalize to two or more straight lines, and
Bae and Chwa [109] treat the general case, i.e., non-isothetic polygonal
networks.
Unfortunately, any slight alteration of the city Voronoi diagram
model, such as non-isothetic networks, individual travel speeds on network
segments, or metrics different from L1 , may yield a diagram size of Θ(n · t),
see [38], and thus significantly increases the computation time and storage
requirement, making the concept less useful in applications.
Time distances also have been used to generalize geometric structures
different from Voronoi diagrams. The convex hull conv(S) of a set S of
n points in the plane is a noteworthy example.
As being the smallest convex set which contains S, conv(S) can be
(implicitly) defined as the minimal set X that contains S and all the line
segments connecting two points in X. Given now, in addition, a straight
line T in the plane (with fast travel speed), the so-called highway hull,
H(S, T ), is the minimal set containing S as well as the quickest paths
Chapter 7. General Spaces & Distances
182
H
Figure 7.31. Highway hull for the Euclidean metric. It consists of convex polygons
along the (horizontal) highway H. Boundary segments touching H are of slope k or −k,
respectively, under which it is best to approach the highway. Slope k increases with the
highway speed. (From [715].)
between all pairs of points in H(S, T ), using the highway T . See Figure 7.31
for an illustration.
Aloupis et al. [49] give an optimal O(n log n) algorithm to find the
highway hull under the L1 -metric, as well as an O(n log2 n) algorithm for the
Euclidean case, improving over earlier O(n2 ) methods. They also identify
the region a highway T must intersect such that the quickest path between
at least one pair of points uses T . Ahn et al. [21] tackle the problem from
the optimization point of view. Given a set S of point sites, and a travel
speed v, find an axis-parallel line T that minimizes the maximum travel
time over all pairs of sites in S. They show that O(n) time algorithms
exist, for both the L1 -metric and the Euclidean metric.
Chapter 8
APPLICATIONS AND RELATIVES
This chapter is devoted to the numerous applications of the Voronoi
diagram V (S) and its dual, the Delaunay triangulation DT(S), in solving
geometric problems. Quite a few applications have already been mentioned
in previous chapters, mostly along with generalizations of the classical
type. Applications in relation to optimization properties of the Delaunay
triangulation have been discussed in Section 4.2.
Whereas distance is implicitly involved in almost all applications of
the Voronoi diagram, we start with some examples of ‘pure’ distance
problems. These distance problems are, apart from their direct use in
practical applications that will be mentioned occasionally, frequently arising
subroutines in more involved geometric algorithms.
8.1. Distance problems
Unless otherwise stated, the following problems address point sites in the
plane under the standard Euclidean distance. However, most solutions can
be generalized, at least to convex distance functions (Section 7.2), and
sometimes also to sites other than points (Chapter 5), and to dimensions
higher than two (Chapter 6).
8.1.1. Post office problem
Our first example is the post office (or nearest neighbor ) problem mentioned
in the seminal paper [637] by Shamos and Hoey. Given a set of n sites (post
offices) in the plane, determine, for an arbitrary query point x, the post
office closest to x.
The Voronoi diagram of the post offices represents the locus approach
to the post office problem: It partitions the plane into regions of equal
answer, i.e., regions whose points x have identical nearest post offices; see
Figure 8.1.
183
184
Chapter 8. Applications and Relatives
s
x
Figure 8.1. Locating a point x among the regions of a Voronoi diagram determines the
post office s closest to x.
It remains to quickly determine the region that contains the query
point, x. This point-location problem has received a lot of interest in
computational geometry. Here we sketch an elementary solution first
mentioned in Dobkin and Lipton [285], called the slab method.
Through each Voronoi vertex a horizontal line is drawn. These extra
lines partition the Voronoi regions into triangles and trapezoids. By
construction, no horizontal ‘slab’ between two consecutive lines contains a
Voronoi vertex in its interior, so that all crossing Voronoi edges are ordered
within the slab. To locate a query point, x = (x1 , x2 ), we use one binary
search for x2 among the slabs, and another one for x1 among the edge
segments of the slab found. This gives us O(log n) query time, at quadratic
storage cost.
There are search structures like in Kirkpatrick [456], Edelsbrunner et al.
[305], and Edelsbrunner and Maurer [308] that are equally efficient, but
need only O(n) storage and can be derived in O(n) time from the Voronoi
diagram. An early method for general subdivisions has been given in Maurer
[523]. With any of the optimal Voronoi diagram algorithms in Chapter 3,
this yields the following solution to the post office problem.
Theorem 8.1. Given a set S of n point sites in the plane, one can, within
O(n log n) time and O(n) storage, construct a data structure that supports
nearest neighbor queries: For an arbitrary query point x, some nearest
neighbor in S can be found in time O(log n).
Storing the history of the Voronoi diagram during incremental insertion
(cf. Section 3.2) even obviates the need of processing the diagram for
8.1. Distance problems
185
point location. Guibas et al. [390] showed that nearest neighbor queries
are supported in (expected) time O(log2 n) by the resulting structure.
Analogously, the order-k Voronoi diagram can be used for finding
the k-nearest neighbors in S of a query point x. Aurenhammer and
Schwarzkopf [102] generalized the practical approach in [390], obtaining
a structure with expected query time O(k log2 n) that additionally allows
insertions and deletions of sites, at a space requirement of O(k(n − k));
see Subsection 9.1.1 for more material on the dynamic post office problem.
A sophisticated technique of compacting order-k Voronoi diagrams in
Aggarwal et al. [18] achieves optimal query time O(k + log n) and space
O(n), but with high constants.
The locus approach also works for more general sites and distance
measures. Let us assume that the site set, S, consists of k non-intersecting
convex polygons with n edges in total. The bisector of two sites is composed
of O(n) many straight or parabolic segments; if we also count the endpoints
of such segments as vertices (of degree 2), then the Voronoi diagram of the
k convex polygons is of complexity Θ(n), as Lemma 5.1 shows. It could be
constructed in time O(n log n) by first computing the line segment Voronoi
diagram of the polygon edges (see Section 5.1), and then joining those
regions that belong to parts of the same convex polygon.
A more efficient structure for solving the post office problem for k
disjoint convex polygons, the so-called compact Voronoi diagram, has been
introduced in McAllister et al. [527]. Using only O(k) many line segments,
it partitions the plane into regions bounded by at most six edges each. With
each region, one or two sites are associated; for any point x in the region,
one of them is the nearest neighbor of x in S.
This planar straight-line graph can be defined quite easily. Its vertices
are the O(k) original Voronoi vertices defined by the k polygonal sites, plus,
for each Voronoi vertex v defined by three sites, the closest points to v on
their respective boundaries. Straight-line edges run from each vertex v to
its closest points and, around the boundary of each site, between the points
closest to its Voronoi vertices; see Figure 8.2. For each site C in extreme
position (i.e., appearing on the convex hull of the set of polygons), a halfline
to infinity is added all of whose points have the same closest point on the
boundary of C.
Theorem 8.2. Given a set S of k disjoint convex polygonal sites in the
plane with a total of n edges, one can, within O(k log n) time and O(k)
storage, construct a data structure that allows nearest neighbor queries to
be answered within time O(log n).
The essential task is in constructing the O(k) many original Voronoi
vertices of the k polygons, spending only O(log ) time on each vertex.
186
Chapter 8. Applications and Relatives
Figure 8.2. A piecewise linear subdivision of size O(k) for solving the post office problem
of convex polygons.
This can be achieved by a clever adaption of the sweep line approach
described in Section 3.4, and by means of a subroutine that computes, in
O(log n) time, the Voronoi vertex of three convex polygons. The subroutine
uses the tentative prune-and-search technique described in Kirkpatrick and
Snoeyink [458]. The same approach works for convex distance functions;
see Section 7.2.
In general dimensions d, the post office problem (which is also called
proximity problem, similarity problem, or nearest neighbor search problem
in the literature) can be efficiently solved by methods different from the
ones described in this subsection. Chapter 10 gives a detailed discussion.
8.1.2. Nearest neighbors and the closest pair
Another distance problem the Voronoi diagram directly solves is the all
nearest neighbor problem: For each point site in the set S, a nearest neighbor
in S is required. An example is shown in Figure 8.3; arrows are pointing
from each site towards its nearest neighbor. For sites in general position, the
resulting nearest neighbor graph is a forest, i.e., a vertex-disjoint collection
8.1. Distance problems
Figure 8.3.
187
The nearest neighbor graph of 11 points.
of trees, each of whose vertices is of outdegree 1 and indegree at most 6.
(The bound on the indegree follows from the containment of this forest in
the minimum spanning tree of S; see Subsection 8.2.1.)
The solution offered by the Voronoi diagram is based on the following
fact.
Lemma 8.1. Let S = P ∪ Q be a disjoint decomposition of the point set S,
and let p0 ∈ P and q0 ∈ Q be such that
d(p0 , q0 ) =
min
p∈P, q∈Q
d(p, q).
Then the Voronoi regions of p0 and q0 are edge-adjacent in V (S).
Proof. Otherwise, the line segment p0 q0 contains a point z that belongs
to the closure of some Voronoi region VR(r, S), where r = p0 , q0 . Let us
assume that r belongs to Q; the case r ∈ P is symmetric. From z ∈ VR(r, S)
follows d(z, r) ≤ d(z, q0 ), hence
d(p0 , r) ≤ d(p0 , z) + d(z, r)
≤ d(p0 , z) + d(z, q0 ) = d(p0 , q0 ) ≤ d(p0 , r)
by minimality of d(p0 , q0 ). Hence, equality must hold, and we obtain
d(p0 , r) = d(p0 , q0 ) and d(z, r) = d(z, q0 ).
But since p0 lies on the bisector B(q0 , r), each point z in the interior
of q0 p0 is strictly closer to q0 than to r, which is a contradiction.
If we apply Lemma 8.1 to the subsets {p} and S\{p}, it follows that
the nearest neighbor of p is sitting in a neighboring Voronoi region (i.e.,
it is a Delaunay neighbor ). Therefore, it is sufficient to inspect, for each
p ∈ S, all neighbors of p in the Voronoi diagram, and select the closest of
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Chapter 8. Applications and Relatives
them. This way, each edge of V (S) will be accessed twice. Since their total
number is linear, due to Lemma 2.3, we can solve the all nearest neighbor
problem by constructing the Voronoi diagram.
Theorem 8.3. Given a set S of n points in the plane, O(n log n) time
and linear space are sufficient for determining, for each p ∈ S, a nearest
neighbor in S.
Once for each p its nearest neighbor in S is known, we can easily
determine a pair of points whose distance is a minimum.
Corollary 8.1. The closest pair among n points in the plane can be
determined within O(n log n) time and linear space.
Hinrichs et al. [407] have shown that it suffices to maintain certain parts
of the Voronoi diagram by a sweep algorithm (cf. Section 3.4), in order to
detect the closest pair.
Whereas finding the closest pair seems an easier task than constructing
the Voronoi diagram, it still has an Ω(n log n) lower bound, by reduction
from the ε-closeness problem; cf. Theorem 3.1.
In dimension d, constructing the Voronoi diagram is no longer the
method of choice for finding the closest pair, due to its exponentially
increasing size; see Section 6.2. Already in 1976, Bentley and Shamos [122]
provided an O(n log n) algorithm for arbitrary but fixed dimensions. Golin
et al. [369] showed that randomization plus use of the floor function (which
is forbidden in the algebraic decision-tree model of computation) allows
the closest pair of n points in d-space to be found within expected O(n)
time, for any fixed d. Additional results on finding closest pairs in high
dimensions, including approximate solutions, are described in Chapter 10.
More involved is the detection of the so-called bichromatic closest pair
in a set S of n points. In this variant each point has a color, either red or
blue, and one asks for the minimum distance between differently colored
points. In the plane, a simple O(n log n) solution exists, by constructing the
Voronoi diagram of, say, the r red points, in O(r log r) time, and performing
point location for the b blue points among the red regions, in O(log r) time
per step (with r + b = n); see Subsection 8.1.1.
Agarwal et al. [11] solve the bichromatic closest pair problem in general
d-space, achieving a runtime of (roughly) O((r · b · log r · log b)2/3 ) in
dimension 3.
The problem of finding the closest pair of n points can be
generalized
to reporting the k closest pairs in S, for some number k ≤ n2 . A bruteforce solution would sort all interpoint distances in time O(n2 log n) and
then report the k smallest. Time O(n2 ) is sufficient if one uses one of
8.1. Distance problems
189
the well-known selection methods [234] for finding, in time O(m), the kth
smallest of m objects (i.e., their so-called k-median). Smid [658] has shown
how to enumerate the O(n2/3 ) smallest distances in time O(n log n) and
linear space.
A simple yet elegant solution using the Delaunay triangulation DT(S)
of S has been provided by Dickerson et al. [272]. Their approach is as
follows.
The closest pair in S corresponds to the shortest edge, e1 , of the
Delaunay triangulation DT(S), as Lemma 8.1 shows. The second-closest
pair, e2 , must be in DT(S) as well, as both edges are, by Kruskal’s
construction, in the minimum spanning tree of S, which is a subgraph
of DT(S); cf. Subsection 8.2.1. (Note that e1 and e2 cannot cross; otherwise,
two of their endpoints would be closer to each other than e2 ’s length, by the
quadrangle inequality. This inequality states that, in any convex quadrangle,
the sum of two opposite sides lengths is strictly smaller than the sum of
the two diagonal lengths.)
But the kth closest pair, (p, q), need not define an edge of DT(S),
not even for small values of k ≥ 3. However, there always exists a path of
Delaunay edges connecting p and q which are of lengths smaller than d(p, q)
each.
This can be used for proving the following elegant algorithm correct.
It maintains a priority queue, Q, of pairs of points by distance. Initially,
Q contains all Delaunay neighbors. In the ith step, the closest pair (p, q)
in Q is removed and reported ith closest. Then, for all Delaunay edges op
satisfying d(o, p) ≤ d(p, q), the pair (o, q) is inserted into Q; similarly, each
pair (p, r) is added to Q where qr is a Delaunay edge of length at most d(p, q).
The performance of this algorithm is stated in the next theorem.
Theorem 8.4. The k closest pairs of a set S of n points can be computed
in time O((n + k) log n) and space O(n + k).
A similar approach even works efficiently in higher dimensions.
Dickerson and Eppstein [273] showed that several interdistance enumeration
problems for a point set S in d-space can be solved by means of DT(S).
The prohibitive size of DT(S) in d-space is reduced by augmenting S to
have a linear-size, bounded-degree Delaunay triangulation, with the method
of Bern et al. [128].
8.1.3. Largest empty and smallest enclosing circle
Suppose someone wants to build a new residence within a given area, as far
away as possible from each of n sources of disturbance — a typical facility
location problem.
190
Chapter 8. Applications and Relatives
x
Figure 8.4. Largest empty circle with center inside the convex hull of the sites. In this
example, the circle is centered at a Voronoi vertex.
If the area is modeled by a convex polygon A over m vertices, and the
disturbing sites by a point set S, we are looking for the largest circle with
center in A that does not contain a point of S. The task of determining this
circle has been named the largest empty circle problem.
Shamos and Hoey [637] observed that this circle must have its center
at a Voronoi vertex of V (S), or at the intersection of a Voronoi edge with
the boundary of A, or at a vertex of A. (See Figure 8.4, where the first case
occurs.) If a circle C with center x ∈ A contains no p ∈ S, not even on its
boundary, we blow it up until it does. If its boundary now contains three
sites then x is a Voronoi vertex; see Lemma 2.1. If there are two sites on
its boundary, x lies on a Voronoi edge. In this case we move x along their
bisector away from the two sites until the expanding circle hits a third site,
or x hits the boundary of A. If, in the beginning, only one point site p ∈ S
lies on the boundary of C then we expand C, while keeping its boundary in
contact with p, until x reaches a vertex of A, or one of the before-mentioned
events occurs.
This observation leads to the following result.
Theorem 8.5. The largest circle not containing a point of S, whose center
lies inside a convex polygon A, can be found within O(m + n log n) time and
linear space. Here, m denotes the number of vertices of A, and n is the size
of S.
8.1. Distance problems
191
Proof. We spend O(n log n) time on constructing the Voronoi diagram
V (S). The largest empty circles centered at the Voronoi vertices can be
determined in constant time each. We can trace the convex boundary of
A through the Voronoi diagram, in order to detect its intersections with
the Voronoi edges, in time O(n + m). Simultaneously we can find, for each
vertex v of A, its nearest neighbor in S, by determining the Voronoi region
containing v.
Based on the above characterization of largest empty circles, Kaplan
and Sharir [443] give an efficient algorithm for preprocessing S, so that,
for a given query point q, we can quickly report the largest (open) disk D
with q ∈ D but D ∩ S = ∅. The storage required by their data structure is
O(n log n), at a preprocessing cost of O(n log2 n), and a query takes time
O(log2 n).
Queries of this kind occur when one wants to transmit a message over
the largest possible area (which is a disk if propagation proceeds uniformly)
such that q will receive the message, but none of the ‘spies’ sitting at the
points of S. A more friendly interpretation is to position a sprinkler to
water the largest possible area that includes plant q, without wettening
any location in S.
Repeatedly selecting centers of largest empty circles (i.e., of optimally
far away points) is a task which also arises in quality mesh generation within
polygonal domains, as we briefly describe next.
Starting with the Delaunay triangulation, DT0 , of a convex polygon P ,
the triangulation can be refined by insertions of centers of largest empty
circles. More precisely, DTi is constructed from DTi−1 by inserting the
center pi of the largest empty circle for DTi−1 within P , for i ≥ 1, until
certain quality criteria are met. (Note that pi might lie on the boundary
of P , thus splitting an edge of P when being inserted.) This idea of
Delaunay refinement (which is also called canonical Voronoi insertion)
originates with Gonzalez [370] and Feder and Green [337], who developed
it for approximating optimal clusterings. Strategies of this kind also play
a role in circle and sphere packing problems; see the survey article by
Fejes Tóth [339].
For mesh generation purposes, Delaunay refinement has been used,
among others, by Chew [213], Ruppert [612], Shewchuk [649], and
Aurenhammer et al. [99]. The last paper discusses the problem of
approximating uniform triangular meshes, in the sense that the maximal
edge length, or the maximal edge ratio or perimeter of a triangle, is to be
minimized. See Figures 8.5 and 8.6 that illustrate a convex polygon, meshed
with small edge ratio. Refinement also works for general simple polygons,
192
Chapter 8. Applications and Relatives
Figure 8.5. Delaunay-refined mesh
with 50 points (from [99]).
Figure 8.6. Further mesh refinement
with 150 additional points.
where the constrained Delaunay triangulation is used (Section 5.4). The
approximation bounds in [99] have been recently improved by Jiang [428],
who confirmed an interesting structural result conjectured there:
For any two triangular meshes T1 and T2 within a fixed simple
polygon P , which have the same number m (but different sets) of internal
mesh vertices, the maximum edge length in T1 is at least the minimum
vertex distance in T2 .
An inverse problem, in some sense, is the smallest enclosing circle
problem. It asks for the circle of minimum radius that encloses a given
set S of n points in the plane. Applications are the placement of a least
powerful transmitter station that can reach a given set of locations, or the
placement of a service station that minimizes the maximum distance to the
customers.
As was observed in Bhattacharya and Toussaint [130], there are two
cases. Either the smallest enclosing circle contains three points of S on its
boundary, or it runs through two antipodal points whose distance equals
the diameter of S, that is, the longest distance spanned by S.
From the convex hull of S its diameter, together with a pair of points
realizing it, can be derived in linear time; then it can be checked if the
smallest circle through these points contains S.
Otherwise, the center v of the smallest enclosing circle C must be
a vertex of the farthest-site (or order n − 1) Voronoi diagram, Vn−1 (S),
introduced in Section 6.5. Namely, the three sites on the boundary of C are
farthest from v.
This yields an O(n log n) algorithm for computing the smallest
enclosing circle. The linear programming technique by Megiddo [531] is
more efficient: For fixed dimension d, it allows the smallest enclosing sphere
8.1. Distance problems
193
for n points in d-space to be constructed in time O(n). A more practical way
of achieving linear (randomized ) time is the elegant minidisk algorithm by
Welzl [702]; here the constant in O has been shown to be a subexponential
function of d by Matoušek et al. [522]. The minidisk algorithm also works
for smallest enclosing ellipsoids.
The smallest enclosing circle problem is the simplest instance of a
clustering problem called k-center problem. This problem asks for a set C
of k centers such that S is covered by k disks centered at C, with the largest
arising disk radius δ getting minimized; see Subsection 8.4.1. In other words,
a partition of S into k subsets (clusters) and a center for each cluster are
sought, such that the maximum distance, δ, of a cluster point to its center
is minimized. Clearly, δ is the radius of the smallest enclosing circle for the
respective cluster.
Efficient algorithms for the k-center problem are partially based on the
observation that clusters are linearly separable hence hull-disjoint. If for
two clusters one point each would lie on the ‘wrong’ side of a straight line,
then re-assignment of points to centers would decrease (or leave equal)
the critical radius δ. Particular attention has been paid to the 2-center
problem, for which various superquadratic algorithms were proposed until
Sharir [640] and Eppstein [322] succeeded in achieving near-linear runtimes;
O(n logc n) for c = O(1), and O(n log2 n), respectively.
Recently, a coloring version of the 2-center problem has been
investigated in Arkin et al. [65]. The points in S are to be colored (pairwise)
either red or blue. Thereby, a given function f (δR , δB ) of the radii of two
disks DR = (cR , δR ) and DB = (cB , δB ) is to be minimized, such that
DR covers all the red points, and DB covers all the blue ones. Unlike
for the bichromatic closest-pair problem (in Subsection 8.1.2), the coloring
is not given but part of the output. This bichromatic 2-center problem
is motivated by an application in air traffic management, where a flight
corridor (the line segment cR cB ) should be designed such that the traffic
outside the corridor is minimized.
For f being the maximum or the sum of radii, polynomial-time solutions
of O(n3 log2 n) and O(n5 polylog n), respectively, are given in [65]. They
also obtain much more efficient algorithms for approximating the optimum
value of f , and for the L∞ -metric.
We have seen that, for the largest empty circle problem, the closest-site
Voronoi diagram V (S) of the input point set S is of help, whereas for the
smallest enclosing circle problem, the farthest-site Voronoi diagram Vn−1 (S)
plays this role. In solving the smallest annulus problem for S, information
from both diagrams is used. Here, two co-centric disks D1 and D2 of radii
r1 > r2 are sought, such that D1 covers S and D2 avoids S, and the width
194
Chapter 8. Applications and Relatives
x
Figure 8.7. Minimum-width annulus covering a set of six points. Its center, x, is located
at a crossing of their closest-site and farthest-site Voronoi diagrams.
r1 − r2 of the annulus D1 \D2 they define is minimized; see Figure 8.7.
Applications stem from testing the roundness of a point set, used as a
tolerance measure in mechanical design; see Le and Lee [489], Swanson
et al. [672], and Smid and Janardan [659].
The center of the smallest annulus is a vertex of V (S), or a vertex
of Vn−1 (S), or the intersection point of two edges in these diagrams. This
gives an O(n2 ) algorithm for its construction, as was first observed in
Ebara et al. [299]. Using different (and more complex) techniques, Agarwal
3
and Sharir improved this to O(n 2 +ε ); we refer the interested reader to
their survey article [16] on geometric optimization algorithms. Moreover, a
(1 + ε)-approximation can be obtained in time O(n + ε−c ), for a suitable
constant c > 1, using the linear programming methods in Chan [188].
If roundness of a convex or simple polygon P is to be checked, then
the medial axis of P (Sections 5.1 and 5.2), and the farthest-site Voronoi
diagram of P ’s vertices, have to be taken for characterizing the location of
the center of the minimum-width annulus that covers P ; see [489, 672].
8.2. Subgraphs of Delaunay triangulations
The Delaunay triangulation DT(S) of a set S of n point sites contains,
as subgraphs, various structures with diverse applications. One example,
the all nearest neighbor graph of S, has already been discussed in
Subsection 8.1.2. Efficient algorithms for computing these structures follow
from the fact that DT(S) has size O(n), and can be constructed in
O(n log n) time, in the plane. This positive effect is partially lost in higher
8.2. Subgraphs of Delaunay triangulations
195
dimensions, as DT(S) may be the complete graph on S already in 3-space.
Alternative methods for computing subgraphs of Delaunay triangulations
in d dimensions, and for approximating them, are discussed in Chapter 10.
8.2.1. Minimum spanning trees and cycles
A minimum spanning tree, MST(S), of S (sometimes also called shortest
connection network for S) is a planar straight line graph on S which is
connected and has minimum total edge length. Such a graph is necessarily
acyclic and thus is a tree. Minimum spanning trees play an important
role, for instance, in transportation problems, pattern recognition, and
clustering. Shamos and Hoey [637] first observed the connection to
Delaunay triangulations.
Lemma 8.2. MST(S) is a subgraph of DT(S).
Proof. Let e be an edge of MST(S), and see Figure 8.8. Removal of e splits
MST(S) into two subtrees, and S into two subsets S1 and S2 . Clearly, e is
the shortest edge connecting S1 and S2 , because a shorter edge, if existent,
would lead to a spanning tree shorter than MST(S). It follows that the
circle C with diameter e is empty of sites in S: A site enclosed by C would
have to belong to either S1 or S2 , leading to a connection between S1 and
S2 shorter than e. By Definition 2.2 in Chapter 2, C proves e Delaunay.
We thus can select the n − 1 edges which build up MST(S)
from the
at most 3n − 6 edges of DT(S), rather than from all the n2 edges spanned
e
Figure 8.8.
Delaunay triangulation, and minimum spanning tree (in bold style).
196
Chapter 8. Applications and Relatives
by S, by standard application of Kruskal’s [475] or Prim’s [597] ‘greedy’
algorithms. Edges are considered in increasing length order, and an edge is
classified as a tree edge if it does not violate acyclicity.
In Kruskal’s version, to detect if a considered edge e = (p, q) yields a
cycle, the forest of subtrees of MST(S) constructed so far (which initially
consists of n singleton vertices, the sites in S) is stored in a union-find data
structure; see e.g. [234, 538]. Performing FIND(p) = i and FIND(q) = j
returns the indices of the sets Mi and Mj that store the vertex sets (subsets
of S) of the current subtrees Ti and Tj that contain p and q, respectively.
Now, if i = j then edge e does not close a cycle but rather joins the subtrees
Ti and Tj into a larger one. In order to reflect this in the data structure,
we join the sets Mi and Mj with UNION(Mi , Mj ).
We choose an implementation that supports the FIND operation in
O(1) time, and all n−1 UNION operations in total time O(n log n). For each
set Mi , we just link all its members (the vertices of Ti ) to the set index, i.
When joining two sets, the links for the smaller set are redirected to the
larger set. This way, each fixed member changes the set which contains it
at most O(log n) times. An overall runtime of O(n log n) for constructing
MST(S) is obtained, including the computation of DT(S).
Lemma 8.2 implies that MST(S) is a crossing-free tree, i.e., its edges do
not intersect in their interiors. This is because the edges in any triangulation
do not cross.
There is a generalization of this lemma, for constrained Delaunay
triangulations and spanning trees. Consider an arbitrary crossing-free
spanning tree, T , of the point set S. We define the minimum spanning
tree of S constrained by T , for short MST|T (S), as the shortest spanning
tree for S that does not cross (but may contain) edges of T . It is easy to
see that MST|T (S) is a subgraph of the constrained Delaunay triangulation
CDT(S, T ) discussed in Section 5.4. Exploiting this containment relation,
Aichholzer et al. [36] showed that the sequence of spanning trees
T0 , T1 , . . . , Tk ,
with T0 = T
and Ti = MST|Ti−1 (S),
for 1 ≤ i ≤ k
terminates at Tk = MST(S), the (unconstrained) minimum spanning tree
of S, after k = O(log n) iterations. This so-called fixed tree theorem allows
us to gradually though quickly transform two arbitrary spanning trees of
a given point set into each other — a result with possible applications to
morphing of skeletons and shapes in the plane.
Lemma 8.2 also generalizes to metrics different from the Euclidean (for
example L1 , see Hwang [416]), and to higher dimensions. When staying in
the plane, equally efficient construction algorithms result.
8.2. Subgraphs of Delaunay triangulations
197
In higher dimensions, however, DT(S) may not be of much use
for the construction of MST(S), because of the possibly quadratic size.
Subquadratic worst-case time algorithms for computing the Euclidean
minimum spanning tree in d-space exist, for example by Yao [710]
and Agarwal et al. [11]. The latter algorithm runs in randomized time
O((n log n)4/3 ) for dimension 3, and uses the computation of a bichromatic
closest pair in S as a subroutine; see Subsection 8.1.2.
It still remains open whether an O(n log n) time algorithm can be
developed, at least for 3-space. The problem is apparently easier for
the L1 -metric and the L∞ -metric. The minimum spanning tree can be
computed in O(n log n) time for these metrics, in fixed dimensions d; see
Krznaric [476].
The minimum spanning tree MST(S) should not be confused with the
(Euclidean) minimal Steiner tree of a point set S, which is the shortest
connection network for S when the usage of extraneous points is allowed for
length reduction. Indeed, already a three-point example shows that adding
a fourth point can reduce the length of the resulting tree structure. In
particular, the Fermat point of a triangle with angles less than 2π
3 , which is
the point minimizing the total distance to the triangle’s vertices, has this
property. (By contrast, the minimum spanning tree would consist of the two
shortest sides of the triangle.) Minimal Steiner trees may use O(n) Fermat
points as extraneous points. Finding a minimal Steiner tree is an NP-hard
problem; see Garey and Johnson [352].
Another related tree concept is the minimum diameter spanning tree
of S, where the Euclidean length of the longest path in the tree is to be
kept to minimum. Figure 8.9 displays an example. This tree is either a
u
p
q
Figure 8.9. The Euclidean minimum diameter spanning tree is a double-star in this
case. Its diameter is realized by a path of 3 edges, drawn in bold style. Note that point u
is connected to p and not to q, whereas d(u, p) > d(u, q).
Chapter 8. Applications and Relatives
198
star or a double-star and — by exploiting its simple topology — can be
found in O(n3 ) time using the farthest-site and the order-(n − 2) Voronoi
diagrams of S (Section 6.5). Consult Ho et al. [408], and Chan [189] for an
O(n17/6+ε ) improvement. One can compute (1 + ε)-length approximations
much faster, in O(n + ε−3 ) time, with the grid-oriented methods in Spriggs
et al. [662]. Some bi-criteria optimization problems based on minimum
diameter spanning trees have been shown to be NP-hard, in Seo et al. [634].
A travelling salesman tour, TST(S), for a set S of point sites in the
plane is a minimum length spanning cycle (cycle passing through all the
sites). It is known that TST(S) is not a subgraph of DT(S), in general.
Dillencourt [276] showed that DT(S) need not even contain any spanning
cycle (a so-called Hamiltonian cycle), and that finding Hamiltonian cycles
in Delaunay triangulations is NP-hard [279]. He also gives a partial
explanation for the fact that DT(S) is Hamiltonian in most cases [277].
A related optimization graph which is known to be not part of DT(S)
is a minimum length matching for S; see Akl [46]. However, DT(S) is
guaranteed to contain some (perfect) matching (i.e., a set of edges that
pairs up all the points in S except possibly one, and hence is of cardinality
n2 ); see Dillencourt [277]. This property is not shared by all triangulations
of S.
Finding a travelling salesman tour has been shown to be NP-hard in
Papadimitriou [578], but a factor-2 length approximation of TST(S) can be
found easily, using MST(S). Let A(S) be a cycle through S that results from
traversing MST(S) in preorder ; consult Figure 8.10. Rosenkrantz et al. [606]
4
5
1
3
6
2
7
8
9
10
11
Figure 8.10.
Minimum spanning tree, and short cycle (in dashed style).
8.2. Subgraphs of Delaunay triangulations
199
observed the following. Let |X| denote the total length of a set X of
edges.
Lemma 8.3. |A(S)| < 2 · |TST(S)|.
Proof. When traversing each edge of MST(S) twice, a tour longer than
A(S) is obtained. Hence |A(S)| < 2 · |MST(S)| holds. To see |MST(S)| <
|TST(S)|, note that removing an edge from TST(S) leaves a path which is
some, but not neccessarily the minimum, spanning tree of S.
A(S) can be constructed in linear time from MST(S). Note that tree
traversal in preorder is facilitated by the fact that the maximum vertex
degree in MST(S) is at most six: Consecutive edges around each tree vertex
form an angle of at least 60◦ , to let them be the two shortest edges of the
spanned triangle.
A more sophisticated construction of a spanning cycle by means of
MST(S) is given in Christofides [222]. An approximation factor of 1.5 is
√
achieved, at an expense of roughly O(n2 n) in construction time.
Another NP-hard problem, which has a factor-2 approximation by
means of MST(S), is the construction of optimal radii graphs; see Chen
and Huang [200]. Let R be a real-valued vector that associates each site
p ∈ S with an individual radius rp > 0. The corresponding radii graph,
GR (S), contains an edge between two sites p, q ∈ S iff d(p, q) ≤ min{rp , rq }.
The optimization problem now asks for a radii vector R∗ such that
GR∗ (S) is connected and |R∗ | = p∈S rp∗ is minimum. Radii graphs have
applications, among others, in the design of strongly connected radio and
sensor networks.
Consider the vector R that takes, as a radius for each site p ∈ S,
the length of the longest edge of MST(S) incident to p. Then GR (S) is
connected, as it has to contain MST(S) as a subgraph. Moreover, we have
the following quality guarantee.
Lemma 8.4. |R| < 2 · |R∗ |.
Proof. Let e = (p, q) be some edge of MST(S). Then e appears at most
twice as a radius in R, namely if e is the longest edge incident to p, and
the longest edge incident to q. This gives |R| ≤ 2 · |MST(S)|.
To verify |MST(S)| < |R∗ |, let us consider GR∗ (S). As being connected,
this graph contains some spanning tree, T . We may orient the edges of T
such that each site p ∈ S (except one) has a unique edge ep of T pointing
at it. As ep is also an edge of GR∗ (S), we have rp∗ ≥ |ep |. Hence |R∗ | >
|T | ≥ |MST(S)|.
Chapter 8. Applications and Relatives
200
Lemma 8.4 remains true for more general measures |R| = p∈S f (rp ),
for any increasing function f . This is because MST(S) also minimizes
2
e∈T f (|e|), for all spanning trees T of S. The case f (r) = r is relevant to
the application mentioned above, as the received power of a radio terminal
decreases with the square of the distance.
8.2.2. α-shapes and shape recovery
Extracting the shape of a given set S of point sites in the plane is a
problem that arises, for example, in data visualization, pattern recognition,
and in particular, in curve reconstruction. To some extent, the shape of S
is reflected by the convex hull of S. The edges of the convex hull are part of
the Delaunay triangulation DT(S). The concept of α-shape, introduced in
Edelsbrunner et al. [306, 300], generalizes the convex hull of S for the sake
of better shape approximation, while still remaining a subgraph of DT(S).
For α > 0, the α-shape α(S) of S is defined to contain an edge between
sites p, q ∈ S iff there is some (open) disk of radius α that avoids S and
has p and q on its boundary. By Definition 2.2 in Chapter 2, α(S) is always
part of DT(S) and thus has only O(n) edges.
For α being sufficiently large, the convex hull of S is obtained. The
smaller is the value of α, the finer is the level of resolution of the shape
of S; see Figure 8.11. The graph α(S) contains no edges (only singleton
points) if the disk diameter, 2α, is smaller than the distance of the closest
pair in S.
The definition of α(S) can be extended to negative values of α,
by requiring that the disks of radius −α fully contain S. In this case,
approximations of the shape of S more crude than the convex hull are
obtained, and the edges of α(S) appear in the dual triangulation of the
farthest-site Voronoi diagram of S, that is, in the farthest-site Delaunay
triangulation of S; see Section 6.5.
α
α
Figure 8.11.
Two α-shapes of a point set for different values of parameter α.
8.2. Subgraphs of Delaunay triangulations
201
It follows that all the edges of the whole family of α-shapes for S, for
−∞ < α < ∞, are contained in two triangulations of size O(n). For each
triangulation edge e, an interval of activity can be specified, containing
all values of α such that α(S) contains e. These activity intervals can be
computed in O(n log n) time, and allow us to extract α(S) for an input
value α in O(n) time; see [306].
Another nice feature of α-shapes is their flexibility. The notion of
α-shape, along with its relationship to Delaunay triangulations, nicely
generalizes to higher dimensions. α-shapes in 3-space are of particular
interest, and implementations have been used in several areas of science
and engineering; see Edelsbrunner and Mücke [307].
Also, the concept of α-shape can be generalized to represent different
levels of detail at different parts of space; see Edelsbrunner [302]. This
is achieved by weighting the sites in the given set S individually. The
corresponding weighted α-shapes are subgraphs of the regular triangulation
for the weighted set S which, in turn, is the dual of the power diagram
for this set. The combinatorial and algorithmic properties of weighted
α-shapes are similar to that of their unweighted counterparts. Interpreting
weighted sites as spheres leads to a compact representation of the
respective union of balls by their weighted α-shapes. This is discussed in
Subsection 6.6.2.
Different approaches for reconstructing shapes and surfaces in 3-space
by means of Delaunay triangulations have been pursued. Boissonnat and
Geiger [143] exploit additional information on the sites available in certain
applications, namely that S is contained in k parallel planes (corresponding
to cross sections of the object to be reconstructed, e.g., in computer
tomography). The Delaunay edges connecting sites in neighboring planes
can be computed in an output-sensitive manner; see Boissonnat et al. [139].
The edges describing the final shape are selected according to several
criteria. Geiger [355] reports that satisfactory shapes are produced even
for complicated medical images.
A particularly useful concept for reconstructing an object from a set S
of sample points on its boundary is the so-called crust of S, introduced
in Amenta and Bern [58]. As does the α-shape, also the crust consists of
faces of the Delaunay triangulation of S. In the two-dimensional case, all
edges are included in the crust that can be enclosed by a circle empty of
sites and of all vertices of the Voronoi diagram of S. In three dimensions,
the (sphere) emptiness condition on Voronoi vertices has to be mildened
for certain reasons, to so-called pole vertices, which are the two vertices per
region (if they exist) that are farthest from the object boundary on either
side. If S fulfills specific sampling conditions (namely, being an r-sample,
202
Chapter 8. Applications and Relatives
a condition based on the medial axis of the object), then the crust indeed
connects S by edges (triangles) in a topologically correct way.
Surface recovery can also be based, as done with advantage in
Boissonnat and Cazals [140], on the natural neighbor interpolants described
in Section 4.2. Manifold reconstruction in high dimensions is a demanding
problem. Various abstract simplicial complexes have been defined and
utilized for this purpose; see Section 10.5 for a short discussion of these
structures, and their relationship to Delaunay simplicial complexes.
An extensive study of curve and surface reconstruction techniques is
provided in the book by Boissonnat and Teillaud [149].
8.2.3. β-skeletons and relatives
Another family of graphs, defined by empty disks and thus consisting of
subgraphs of DT(S), is the family of β-skeletons of a set S of point sites.
β-skeletons have been introduced in Kirkpatrick and Radke [457] as a class
of empty neighborhood graphs and have later received increased attention,
not least because of their relation to minimum-weight triangulations.
In [457] both a circle-based and a lune-based version of β-skeletons are
proposed. Here we will mainly consider the former, as defined below.
In contrast to α-shapes, the radii of the empty disks defining a
β-skeleton are not fixed but depend on the interpoint distances in S. Let
p, q ∈ S. For β ≥ 1, an edge between p and q is included in the β-skeleton
β(S) iff the two disks of diameter β · d(p, q) which have the segment pq as
a chord are empty of sites in S. See Figure 8.12.
Note that β(S) is a subgraph of β (S) if β > β . For the minimum value,
β = 1, the two disks above coincide, and pq becomes their diametrical chord.
The so-called Gabriel graph of S is obtained, which thus contains all possible
circle-based β-skeletons for S. Whereas β(S) may be disconnected for any
β > 1, the Gabriel graph is always connected, as it contains the minimum
spanning tree of S as a subgraph; cf. the proof of Lemma 8.2. It is easy to
see that the Gabriel graph just consists of those edges of DT(S) that cross
their dual Voronoi edges. This observation yields an O(n log n) algorithm for
its construction. In fact, also general β-skeletons can be constructed from
DT(S) in linear time; see Jaromczyk et al. [427] and Lingas [508]. Gabriel
graphs have proved useful in the processing of geographical data, see Gabriel
and Sokal [346] and Matula and Sokal [519], and more recently, in graph
drawing questions and for routing in mobile networks, see Section 8.3 and
Subsection 9.4.2.
It has been first observed by Keil [449] that, for β large enough, β(S)
is contained in the minimum-weight triangulation, MWT(S), of S, which
8.2. Subgraphs of Delaunay triangulations
Figure 8.12.
Disconnected (circle-based) β-skeleton for β =
203
6
.
5
is defined as a triangulation of S having√minimum total edge length; see
also Section 4.2. The original bound β ≥ 2 in [449] has been improved
√ in
Cheng and Xu [206] to β > 1.1768, which is close to the value 2/ 3 for
which a counterexample is known. Apart from the edges of the convex hull
of S, only the shortest edge in S, as well as so-called unavoidable edges have
been known to be part of MWT(S) before. (An edge interior to the convex
hull of S is called unavoidable if no other edge spanned by S crosses it.
Unavoidable edges have to appear in every triangulation of S.)
One of the heuristics for the intriguing problem of constructing
MWT(S) first computes β(S) for the smallest admissible value of β. The
obtained k connected components of β(S) then can be completed lengthoptimally to a triangulation, using dynamic programming, in O(nk+2 ) time;
see Cheng et al. [204]. Thus, if β(S) is connected (k = 1) or has only
a constant number of components, then MWT(S) can be constructed in
polynomial time. Unfortunately, k tends to be linear in n.
Let us mention that there are more effective approaches to constructing
MWT(S). One is based on the notion of a light edge, which is an edge e
spanned by S such that all other spanned edges that cross e are longer.
If the set of light edges for S happens to form a full triangulation of S,
then it has to be the minimum-weight triangulation and, at the same time,
the greedy triangulation (Section 4.2). Otherwise, a partition of the greedy
204
Chapter 8. Applications and Relatives
triangulation into k ≥ 2 edge subsets is induced, and if k = O(1) then
such k-level greedy triangulations give a constant-factor approximation of
MWT(S); see Aichholzer et al. [32, 39].
Interestingly, there is a graph that ‘almost always’ coincides with
MWT(S): the so-called LMT-skeleton defined in Belleville et al. [119] and in
Dickerson and Montague [274]. This is an easy-to-compute large subgraph
of the intersection of all locally minimal triangulations (LMTs) that exist
for S (cf. Section 4.2). For more information on LMT-skeletons, in particular
concerning their fast construction, the interested reader may consult the
survey article on optimal triangulations by Aurenhammer and Xu [106].
A slight modification of the empty neighborhood of an edge in S
gives rise to the so-called relative neighborhood graph of S. For p, q ∈ S,
inclusion of the edge pq into this graph is done iff the intersection of
the two disks Dp and Dq , centered at p and q and of radius d(p, q), is
empty of other sites. Note that the relative neighborhood graph is always
a subgraph of the Gabriel graph, as the intersection Dp ∩ Dq (the socalled lune) covers the diametrical disk of pq. On the other hand, this
graph is still connected, as it has to contain the minimum spanning tree
of S: Every edge e of MST(S) has an empty lune L(e), because s ∈ L(e)
for another site s would mean that e is the longest side of the triangle
spanned by s and e, which cannot be in MST(S). Applications in pattern
recognition are reported in Toussaint [686]. Relative neighborhood graphs
are constructible in O(n log n) time by exploiting — once more — their
containment in DT(S); see Supowit [670] and Kirkpatrick and Radke [457].
When we let the disks Dp and Dq move on the straight line through p
and q, such that their radii enlarge to β · d(p.q) for β > 1 while their
boundaries still touch the segment pq at one endpoint, we obtain the
spectrum of lune-based β-skeletons mentioned at the beginning of this
subsection.
Again, these skeletons get sparser with increasing values of β; they are
all part of the relative neighborhood graph. For β = ∞, the lune Dp ∩ Dq
degenerates to a strip orthogonal to the segment pq, and the ∞-skeleton
is acyclic and consists of a forest of monotone paths. In spite of being
subgraphs of DT(S) — as are their circle-based analogs — lune-based
β-skeletons are less easy to deal with algorithmically. For 1 < β ≤ 2,
a runtime of O(n log n) can still be achieved; see e.g. Lingas [508]. Also,
Majewska and Kowaluk [515] showed recently that the entire ‘β-spectrum’
within this range is computable in O(n log2 n) time. The best known
construction algorithm for 2 < β ≤ ∞ is by Kowaluk [474] and runs in
time O(n3/2 log1/2 n), which slightly improves over an earlier solution by
Rao and Mukhopadhyay [601].
8.2. Subgraphs of Delaunay triangulations
205
Versions of β-skeletons for values β < 1 exist, too. These graphs are
non-planar in general, and get arbitrarily dense in the extreme case: The
complete graph Kn is obtained, once β falls short of half of the minimum
interpoint distance in S. Hurtado et al. [413] gave worst-case optimal O(n2 )
time algorithms for these structures.
8.2.4. Paths and spanners
Let S be a set of n point sites in the plane. A (connected) straight line
graph G on S is said to have dilation t if, for any p, q ∈ S, the length of
the shortest path in G between p and q is at most t · d(p, q), where d stands
for the Euclidean distance. In this case, G is also called a t-spanner (of the
complete Euclidean graph) for S.
Intuitively speaking, spanners are graphs on the point set S which
contain shortest paths not much longer than the direct distance between
source and target, and the objective is to keep their number of edges small,
i.e., significantly subquadratic. Sparse t-spanners (i.e., which have only
O(n) edges) are of special interest in various applications, including the
field of robotics and the study of wireless ad hoc networks. The dilation t
is also referred to as the spanning ratio or the stretch factor of a t-spanner
in the literature.
The minimum spanning tree MST(S), though being optimally sparse,
may have dilation Θ(n). Also the (circle-based) β-skeleton can have
unbounded dilation, including the special case of Gabriel graphs; see
Eppstein [323]. By contrast, DT(S) is sparse but ‘sufficiently connected’ to
exhibit constant dilation, independently from the size n of S. Consequently,
good spanners for S can be constructed in O(n log n) time. A dilation of
t ≈ 5 for DT(S) is proved in Dobkin et al. [283], a result which has been
strengthened to t ≈ 2.5 in Keil and Gutwin [450] and, very recently, to
t < 1.998 in Xia [708]. An easy lower bound is t ≥ π/2. However, as Bose
et al. [159] have shown, the dilation of DT(S) is almost always strictly larger
than π2 . In certain applications, sparse spanner graphs where the degree of
every vertex is bounded by a constant are sought. In fact, DT(S) contains
such a graph; see, e.g. Li and Wang [506] and Bose, Smid, and Xu [161].
Surprisingly, competing upper bounds on the dilation t can be achieved
by taking the Delaunay dual of generalized Voronoi diagrams.
The convex distance function whose unit circle is an equilateral triangle
leads to a shape Delaunay tessellation with t = 2. (See Section 7.2 for convex
distance functions, and Subsection 7.2.2 for their dual shape tessellations.)
This bound was shown in Chew [212], along with the following result
concerning the constrained Delaunay triangulation (Section 5.4) for the
206
Chapter 8. Applications and Relatives
L1 -metric:
√ There is always a path between two given sites, whose length is
at most 10 times the geodesic distance with respect to the constraining
line segments. This yields an O(n log n) time algorithm for computing
constant approximations of optimal paths in the presence of polygonal
obstacles. It can be shown that shape Delaunay tessellations always exhibit
a constant dilation, that depends only on the underlying convex shape; see
Bose et al. [156].
Path lengths can be reduced by overlaying several graphs that show
a small dilation in pre-specified directions. This idea has been pursued
by Abam et al. [1], who utilize shape Delaunay tessellations induced by
rotational copies of a sharp diamond whose smaller angle is a function of
some (small) value ε > 0 (Figure 8.13). For directions close to parallel to
the diameter of the diamond, the dilation then is only 1 + ε. Now, using
O(1/ε2 ) symmetrically rotated diamond copies, the same dilation can be
maintained overall, at a price of increasing the size of the spanner graph
to O(n/ε2 ). Moreover, a kinetic (1 + ε)-spanner is obtained in this way,
which yields roughly O(n2 /ε2 ) events in the worst case, and has only O(1)
response time, assuming that the trajectories of the points in S can be
described by reasonably simple curves; cf. Section 9.1.
Alternative triangulations of S are known to exhibit good spanner
properties. Examples are the greedy triangulation and the minimum-weight
triangulation; see Das and Joseph [238]. But only for DT(S) and its
convex shape variants there are algorithms available that are worst-case
efficient and easy to implement. However, graphs based on certain edge
exclusion properties (which are shared, for example, by the minimumweight triangulation) lead to good and efficiently computable spanners as
well [161]. Observe that Delaunay triangulations, by contrast, are defined
via an edge inclusion property, namely, the empty-circle (or empty-shape)
property. Sharpening this inclusion property led to various subgraphs of
DT(S), like the Gabriel graph and the nearest neighborhood graph, in
Subsection 8.2.3.
In higher dimensions, Delaunay tessellations may lose their sparseness
and thus are of minor interest for computing spanners. Different techniques
have been used with success; see e.g. Vaidya [688], Chandra et al. [192], and
the methods described in Chapter 10. An extensive treatment of spanner
graphs is presented in the book by Narasimhan and Smid [561].
Let us mention that, in principle, any convex partition with vertex set S
(i.e., any partition of the convex hull of S into convex polygons whose edges
cover the points in S) is a candidate for a spanner graph — for instance,
for the purpose of compass routing of messages, where network links are
followed so as to get closer to the final recipient at each particular step.
8.3. Supergraphs of Delaunay triangulations
207
r
p
q
s
Figure 8.13. Shape Delaunay tessellations for a fixed set of sites and two different
diamond shapes. The graphs allow for short paths in the horizontal and vertical direction,
respectively.
Convexity ensures that messages can never get stuck at any node of the
network.
There exists a considerable literature on minimum convex partitions,
concerning their size (number of faces), weight (total edge length), and
other criteria to be optimized, also in higher dimensions. We refer to
Dumitrescu et al. [291], Dumitrescu and Tóth [292], and the references
given there. In particular, minimum-weight convex Steiner partitions in the
plane, where extraneous points are admitted to reduce weight, and points
of S might lie on edges of the partition, are of relevance to spanner graph
extraction. In [292] it is shown that such a partition can never be longer
than O(W · log n/ log log n), where W denotes the weight of a minimum
spanning tree of S.
8.3. Supergraphs of Delaunay triangulations
The Delaunay triangulation is a planar graph and thus has a linear number
of edges and triangles. In spite of these desirable features, more dense (nonplanar) graphs which still exhibit some of the Delaunay properties, like wellshaped triangles, are sought in certain applications. In particular, several
useful supergraphs of DT(S) have been investigated in the literature.
8.3.1. Higher-order Delaunay graphs
The Delaunay triangulation can be extended to higher order, in a way quite
similar to the Voronoi diagram (Section 6.5). For k ≥ 0, Gudmundsson
et al. [386] define the order-k Delaunay graph, k-DT(S), of S to contain
an edge between two sites p, q ∈ S if and only if there exists a circle
passing through p and q that encloses at most k sites of S; see Figure 8.14.
Equivalently, one can define the scope of a triangle spanned by S as the
208
Chapter 8. Applications and Relatives
Figure 8.14. Delaunay triangulation DT(S) (left), and additional edges of the Delaunay
graphs 1-DT(S) (middle) and 2-DT(S) (right). 2-DT(S) is already the complete
geometric graph K7 on these seven sites.
number of sites enclosed by its circumcircle. The graph k-DT(S) then
contains all triangles of scope at most k.
Clearly, 0-DT(S) is just the classical Delaunay triangulation DT(S),
and we have the inclusion property k-DT(S) ⊂ (k + 1)-DT(S). The
motivation behind the generalization to higher order stems from the wish
of gaining more flexibilty in choosing triangles, to build triangulations that
model elevation terrains in geographic information systems applications.
However, there exist edges and triangles in the graph k-DT(S) which cannot
be used in any triangulation of S built from triangles of scope k. In [386]
it is shown how to compute all the (in this sense) useful triangles in time
O(nk 2 + n log n). They also prove that every triangulation of S with scope-1
triangles can be obtained from DT(S) by flipping independent edges, i.e.,
edges that do not belong to the same triangle.
Triangulations of S which are solely composed of triangles of scope ≤ k
are called order-k Delaunay triangulations. The family of order-1 Delaunay
triangulations is a suitable class where optimization regarding additional
criteria can be done efficiently. Assume that some third coordinate is
given for each point site in S, and consider the resulting triangular
surface in three-space where sites are elevated accordingly. Then an order-1
triangulation that minimizes the number of local extrema in its terrain can
be computed in O(n log n) time. Also, shapes of triangles and degrees of
vertices can be optimized efficiently within that class. The maximal size
of this class is 2n−3 ; see Mitsche et al. [545], who also stress the fact
that arbitrary large point sets may have only a single order-k Delaunay
triangulation, even for high k (which then is just DT(S)).
Of particular interest in terrain modeling are constrained versions of
low-order Delaunay triangulations, studied in Gudmundsson et al. [387].
(See Section 5.4 for the concept of a constrained Delaunay triangulation.)
In this setting, a constraining set L of edges is prescribed, and a Delaunay
triangulation of lowest order is sought which includes L. Note that, by
8.3. Supergraphs of Delaunay triangulations
209
contrast, the (classical) constrained Delaunay triangulation for S and L
may contain triangles of very high scope. Define the useful order of an edge
e ∈ L as the lowest order of a triangulation of S that includes e. Then,
if all edges in L have a useful order k ≤ 3, the lowest-order completion
to a triangulation which includes L can be computed in O(n log n) time.
For k ≥ 4, no polynomial-time algorithm is known for that problem.
Several related variants of constrained order-k Delaunay triangulations are
discussed in Silveira and van Krefeld [655].
On the combinatorial side, Abellanas et al. [4] give an upper bound of
3(k + 1)(n − k − 2) on the number of edges of k-DT(S). They also study
some graph-theoretic properties of k-DT(S), proving, among others, that
the order-15 Delaunay graph always has to contain some Hamiltonian cycle.
Recall from Subsection 8.2.1 that, by contrast, DT(S) is not Hamiltonian,
in general. The conjecture is that order 1 already suffices to restore this
property.
Moreover, they show that any two order-1 Delaunay triangulations of a
given point set S can be transformed into each other by flips that generate
only triangulations in that class. That is, the flip graph of all order-1
Delaunay triangulations of S is connected; cf. Section 6.3. Connectivity
is lost for k ≥ 2, in general, unless the point set S is in convex position.
More results on k-DT(S) and its subgraph, the order-k Gabriel graph,
are given in Bose et al. [157], concerning diameter, connectivity, dilation,
and other properties. For example, the dilation of the latter graph is Θ( nk )
√
for k ≥ 1 in the worst case, which generalizes the bound of Θ( n) for the
classical (order-0) Gabriel graph (cf. Subsections 8.2.3 and 8.2.4).
Ábrego et al. [8] provide bounds on the number of crossings, both for
the minimum and maximum number attained when all sets S of n points
are considered. k-DT(S) realizes Ω(k 3 n) crossings for every point set S in
general position, and O(k 2 n2 ) crossings in the worst case, thus relatively
few compared to Θ(n4 ) for the complete graph Kn . These numbers, and
corresponding (smaller) numbers for several subgraphs of k-DT(S) given
in [8], might be interesting from the point of view of graph drawing when
proximity information is to be preserved.
Notice that an edge, pq, is in the graph k-DT(S) iff there are two
adjacent regions R and R in the order-(k + 1) Voronoi diagram Vk+1 (S)
of S, such that p is among the k+1 sites defining R, and q is among the k+1
sites defining R : Each vertex of Vk+1 (S) is the center of a circle through
three sites and with k or k − 1 sites inside, hence the spanned triangle is of
scope at most k. Still, the dual of Vk+1 (S) for k ≥ 1 is not k-DT(S), but
rather the centroid-based convex hull projection in Subsection 6.5.3, which
is a planar graph.
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Chapter 8. Applications and Relatives
8.3.2. Witness Delaunay graphs
Another type of graphs related to DT(S) has been investigated recently.
The so-called witness Delaunay graph of S with respect to a (witness) point
set W , for short DT(S, W ), includes an edge between two points p, q ∈ S
whenever there exists a circle that passes through p and q but does not
enclose any point from W .
Figure 8.15 gives an illustration. If W = ∅, then DT(S, W ) is just
the complete graph on S. On the other hand, if W = S, then we have
DT(S, W ) = DT(S). So the point set W can be used to control the
inclusion of edges into this graph. Similar concepts, though based on
interpoint distances rather than on empty circle (or sphere) properties,
are the witness complexes discussed in Section 10.5. These are abstract
simplicial complexes, which are related to Delaunay simplicial complexes in
a certain sense, and have been applied to manifold reconstruction problems
in high dimensions.
Dillencourt [278] proved that all maximal outerplanar graphs (i.e.,
planar graphs all of whose bounded faces are adjacent to the unbounded
face) are realizable as subgraphs of Delaunay triangulations. Inspired by
that result, Aronov et al. [70] study witness Delaunay graphs concerning
their ability of realizing other graph classes. For example, all trees can
be realized, whereas non-planar bipartite graphs cannot, regardless of the
choice of S and W . (A graph is called bipartite if its vertex set can be
partitioned into two subsets such that no edges run within either subset.)
Figure 8.15. A tree realized as a witness Delaunay graph. For all non-tree edges
(dashed) between sites (•), there is no enclosing circle that does not contain some witness
point ().
8.4. Geometric clustering
211
They also show that DT(S, W ) can be computed in time O(n log2 n +
k log n), where k denotes the (potentially superlinear) number of edges of
DT(S, W ), and |S| = |W | = n.
In graph drawing applications, witness-controlled versions of the
Gabriel graph (Subsection 8.2.3) have been considered. The witness Gabriel
graph of S and W consists of all edges spanned by S whose diametrical circle
encloses no points from W .
Whereas all biconnected outerplanar graphs admit drawings as
(classical) Gabriel graphs, trees that have vertices of degree higher than
five cannot be drawn in that way; see Lenhart and Liotta [503] and Bose
et al. [160], respectively. However, all trees become drawable when witness
Gabriel graphs are used. This was shown by Aronov et al. [71] who also
discuss several other properties of this class of geometric graphs.
8.4. Geometric clustering
Clustering a set of data means finding subsets (called clusters) whose inclass members are similar, and whose cross-class members are dissimilar,
according to a predefined similarity measure. Data clustering is important
in diverse areas of science, and various techniques have been developed; see
e.g. Hartigan [397]. In many situations, similarity of objects has a geometric
interpretation in terms of distances between points in d-space.
The role of Voronoi diagrams in the context of clustering is manifold.
For certain applications, the relevant cluster structure among the objects is
well reflected, in a direct manner, by the structure of the Voronoi diagram of
the corresponding point sites; see e.g. Ahuja [24]. For instance, dense subsets
of sites give rise to Voronoi regions of small area (or volume). Regions of
sites in a homogeneous cluster will have similar shape. For clusters having
a direction-sensitive density, the regions will exhibit an extreme diameter
in the corresponding direction.
Perhaps more important is the fact that numerous types of optimal
clusterings are induced by Voronoi diagrams, and/or can be computed by
first computing a Voronoi diagram, for a certain set of sites. This set need
not coincide with the set of points to be clustered.
A general distinction is between offline and online clustering methods.
The former methods assume the availability of the whole data set before
starting, whereas the latter cluster the data upon arrival, according to an
online classification rule. Though Voronoi diagrams play a role in online
clustering, too, see e.g. Hertz et al. [405], we restrict attention to offline
methods below.
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Chapter 8. Applications and Relatives
8.4.1. Partitional clustering
Let X be a set of n points in d-space. A k-clustering of X is a partition
of X into k subsets C1 , . . . , Ck , called clusters. The dissimilarity of a
single cluster C is measured by an appropriate intra-cluster criterion
µ(C), which may be variance, diameter, radius, etc. The dissimilarity
of the whole clustering, in turn, is expressed as a (usually monotone)
function f of µ(C1 ), . . . , µ(Ck ), called the inter-cluster criterion. Common
examples for f are the maximum or the sum. C1 , . . . , Ck is called optimal
if f (µ(C1 ), . . . , µ(Ck )) is minimal for all possible k-clusterings of X.
If k is part of the input, the problem of finding an optimal k-clustering
is NP-hard in general, even in the plane; see Capoyleas et al. [182] for
references. For fixed k, polynomial-time algorithms are known for various
criteria. This is due to the fact that optimal clusters are separable in a
certain sense, and in several cases are induced by the regions of (generalized)
Voronoi diagrams. This approach was first systematically used in [182].
A common intra-cluster criterion is variance,
µ(C) =
1 d(p, q)2 .
|C|
p,q∈C
Variance can be rewritten as p∈C d(p, s(C))2 , with s(C) being the centroid
(center of mass) of C. Note that s(C) is the point s∗ in d-space that
minimizes the sum of squared distances, p∈C d(p, s∗ )2 , to C. Hence, if
the inter-cluster criterion f is sum, the clustering problem is equivalent
to the k-centroid problem: Given a set X of points to be clustered, find
a set S of k sites (cluster centroids) that minimizes p∈X d(p, S)2 , where
d(p, S) is the distance from p to the closest site in S. This implies that the
optimal k-clustering C1∗ , . . . , Ck∗ (which is also called k-means clustering in
the literature) has the following nice property; see Boros and Hammer [155]:
For 1 ≤ i ≤ k, cluster Ci∗ is contained in the region of s(Ci∗ ) in the Voronoi
diagram of S = {s(C1∗ ), . . . , s(Ck∗ )}.
The candidates for an optimal k-clustering of X thus are the possible
partitions of X induced by a Voronoi diagram of k point sites. The
number of such partitions is nO(dk) in d-space; see Inaba et al. [421]. This
implies a polynomial-time algorithm for computing an optimal k-clustering
for fixed k, which proceeds by enumerating all candidate solutions and
comparing their dissimilarities.
The problem becomes considerably easier when the set S of k ‘cluster
centers’ is fixed in advance. A clustering of X that minimizes the sum of
the squared distances of the clusters to their centers is easily found by just
constructing the Voronoi diagram of S; its regions will induce the desired
8.4. Geometric clustering
213
optimal partition of X. Of more interest is the case where the sizes of the
k clusters are prescribed too. Now, a corresponding least-squares clustering
is induced by a power diagram of S (Section 6.4), a fact which leads to an
algorithm with roughly O(k 2 n) runtime, if X is a set of n points in the
plane; see Aurenhammer et al. [97].
In Inaba et al. [421], optimal k-clusterings for the intra-cluster measure
d(p, q)2
µ(C) =
p,q∈C
(like variance, but without division by |C|) are considered. Again, these
clusterings are induced by a kind of power diagram of the cluster centroids,
weighted additively by |C| and multiplicatively by µ(C); cf. Section 7.4 for
mixed weighted distances.
The reason why sums of squared distances, rather than just sums of
distances, are used in the definition of the above intra-cluster measures
partially lies in the intrinsic difficulty of solving the following, seemingly
easy, question: Given a cluster C of size m ≥ 4, find a point (cluster
center) F that minimizes
d(p, F ).
p∈C
F is called the Fermat point (or Fermat-Weber point ) of C. Its exact
location is the solution to a high-degree polynomial in the coordinates,
and can be computed only by numerical algorithms; see Bajaj [110]. For
m = 3, a classical exact construction for F exists.
Another popular measure µ(C) is the radius of the smallest sphere
enclosing a cluster C. If maximum is taken as the inter-cluster criterion,
the k-center problem is obtained: Find a set S of k centers for X, such that
rS = max d(p, S)
p∈X
is minimized. In other words, choose a set S of centers such that X can be
covered by k congruent spheres of minimum radius rS . (This simplifies to
the smallest enclosing sphere problem for k = 1; see Subsection 8.1.3). It
is easy to see that the Voronoi diagram of S gives rise to optimal clusters.
Capoyleas et al. [182] observed that an optimal k-clustering for µ, for
any monotone increasing inter-cluster criterion f , is induced by the power
diagram of the enclosing spheres.
For both intra-cluster measures µ above (but not for the Fermat point
case), polynomial-time algorithms for fixed k result from considering the
nO(dk) candidate clusterings.
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Chapter 8. Applications and Relatives
For k = 2, the ‘Voronoi property’ of the 2-centroid or 2-center problem
just means linear separability of the two optimal clusters. (Note that linear
separability of two sets is equivalent to their (convex) hull-disjointness.)
However, linear separability for all pairs of clusters in a given clustering does
not always imply that this clustering has a realization by means of Voronoi
or power diagrams. An example is the intra-cluster criterion diameter in
2-space [182].
The problem of constructing linearly (or circularly) separable
clusterings in the plane, where each cluster C of X is separable by a straight
line (or circle) from X \ C, is addressed in Dehne and Noltemeier [251] and
Heusinger and Noltemeier [406]. They give a polynomial-time algorithm for
deciding the existence, and finding an optimal clustering, with prescribed
cluster sizes. A correspondence between separable clusters and regions of
higher-order Voronoi diagrams for X (Section 6.5) is exploited.
The order-k Voronoi diagram, Vk (X), of X is also useful for selecting
from X a k-sized cluster C ∗ of minimal dissimilarity µ(C ∗ ). This approach
is pursued in Boyce et al. [162] and in Aggarwal et al. [19]. For instance,
if µ is variance, C ∗ has a non-empty region in Vk (X). If µ is diameter,
C ∗ is contained in a subset of X having a non-empty region in V3k−3 (X).
These properties hold in arbitrary d-space, and lead to efficient cluster
selection algorithms in the plane. Moreover, if µ(C) measures the perimeter
of the axis-aligned enclosing square, or rectangle, of a cluster C, higherorder Voronoi diagrams in the L∞ -metric (cf. Sections 6.5 and 7.2) yield
improved solutions. Except for the diameter case, the running time of all
these algorithms is dominated by the cost of computing an order-k diagram.
8.4.2. Hierarchical clustering
Hierarchical methods are based solely on a given inter-cluster distance δ.
They cluster a set S of n points as follows. Initially, each point is considered
to be a cluster itself. As long as there are two or more clusters, a pair C, C of clusters is joined into one cluster if δ(C, C ) is minimum for all cluster
pairs.
A frequently used inter-cluster distance is the single-linkage distance
δ(C, C ) = min{d(p, q) | p ∈ C, q ∈ C }.
As was pointed out in Shamos and Hoey [637], constructing the singlelinkage cluster hierarchy for S just means simulating the greedy algorithm
of Kruskal [475] for computing the minimum spanning tree MST(S) of S.
We thus obtain a time bound of O(n log n) in the plane, and subquadratic
bounds in d-space; see Subsection 8.2.1.
8.4. Geometric clustering
215
Figure 8.16. Removing the three longest edges of MST(S) leaves a 4-clustering of S
where the minimum distance arising between any two clusters is maximized.
In fact, after having performed n−k steps in constructing the hierarchy
(that is, after having added the n − k shortest edges of MST(S)), the
intermediate k-clustering C1 , . . . , Ck is optimal in the following sense;
see Asano et al. [75]: It maximizes the minimum single-linkage distance
between the clusters, for all possible k-clusterings of S. Figure 8.16
illustrates these observations. Note further that C1 , . . . , Ck are hull-disjoint,
by construction.
The practical value of single-linkage clustering is, however, restricted
by the fact that the produced clusters tend to exhibit large dissimilarity in
terms of variance, radius, and diameter. To remedy this deficiency, other
inter-cluster distances have been used. Among them is the complete-linkage
distance
δ(C, C ) = max{d(p, q) | p ∈ C, q ∈ C }.
Constructing the complete-linkage hierarchy for S efficiently is much more
elusive. Figure 8.17 depicts a planar example. A direct approach leads to an
O(n3 ) time and O(n) space algorithm in general d-space. (The frequently
cited O(n2 ) time insertion algorithm in Defays [246] only approximates the
hierarchy. Its output depends on the insertion order.)
In the plane, it seems of advantage to use, as an auxiliary structure,
the Voronoi diagram of the intermediate clusters C1 , . . . , Ck , defined by
the Hausdorff distance h(x, C) = max{d(x, p) | p ∈ C} of a point x to a
cluster C; see Section 6.5. Unfortunately, the closest pair of clusters does
not necessarily yield adjacent regions in this diagram. Moreover, there exist
examples where the clusters violate hull-disjointness.
216
Chapter 8. Applications and Relatives
5
6
9
3
8
1
2
4
7
10
Figure 8.17. Complete-linkage hierarchy for the same point set as in Figure 8.16. In
the resulting (self-crossing) spanning tree, the edges joining the clusters are numbered
in their order of generation. Note that the tree is pointed, in the sense that the edges per
vertex span an angle less than π (in fact, less than π2 ).
An O(n log2 n) time complete-linkage clustering algorithm for a set S
of n points in the plane has been given in Krznaric and Levcopoulos [477].
Among other structures, they use the Delaunay triangulation DT(S),
and the farthest-site Voronoi diagram (Section 6.5) of the points in the
intermediate clusters Ci . They also show that approximations of the
complete-linkage hierarchy for S can be obtained from DT(S) in O(n) time.
Clustering data in higher dimensions is intricate. An interesting and
effective approach is so-called topological clustering, based on abstract
simplicial complexes related to Delaunay complexes; see Section 10.5. In
particular, the results in Ghrist and Muhammad [362] and de Silva and
Carlsson [259] are relevant to applications like data visualization, sensor
networks, and others.
8.5. Motion planning
The general motion planning problem (also called path planning problem)
is in finding a trajectory for a robot, from a given start configuration
to a target configuration, that avoids collision with a set of obstacles
and possibly satisfies additional requirements concerning cost, speed, or
other constraints. Applications range from industrial robots to autonomous
vehicles, and on to character motions in game design.
There is a rich literature on both the algorithmic aspects of this task
and their practical realization. The reader may consult, e.g., the early
surveys by Alt and Yap [52, 53], Yap [711], the handbook chapter by
Sharir [641], or the monographs by LaValle [485] and Latombe [484].
8.5. Motion planning
217
In the following we will present some aspects of the general motion
planning problem that involve Voronoi diagrams and employ methods
interesting in their own right.
Of relevance in this context are also the collision detection methods
in Kirkpatrick et al. [459], Abam et al. [2], and others; consult the survey
article by Lin and Manocha [507]. The first paper uses pseudo-triangulations
(see Section 6.3) of free space as an underlying data structure. As these
methods are not primarily based on Voronoi diagrams, we refrained from
their detailed description, though.
8.5.1. Retraction
Suppose that for a disk-shaped robot centered at some start point, s, a motion
to some target point, t, must be planned in the presence of n line segments as
obstacles. We assume that the line segments are pairwise disjoint, and that
there are four line segments enclosing the scene (Figure 8.18).
While the robot is navigating through a gap between two line segments,
1 and 2 , at each position x its ‘clearance’, i.e., its distance
d(x, i ) = min{d(x, y) | y ∈ i }
to the obstacles, should be maximized. This goal is achieved if the robot
maintains the same distance to either segment. In other words, the robot
should follow the bisector B(1 , 2 ) of the line segments 1 and 2 until its
distance to another obstacle gets smaller than d(x, i ).
t’
t
z(t)
l(t)
s’
s
z(s)
Figure 8.18. Moving a disk from s to t in the presence of barriers is facilitated by their
Voronoi diagram.
218
Chapter 8. Applications and Relatives
Roughly, this observation implies that the robot should walk along the
edges of the Voronoi diagram V (S) of the line segments in S = {1 , . . . , n }
(see Sections 5.1 and 5.5). This diagram is connected, due to the four
surrounding line segments.
If start and target point are both lying on Voronoi edges, the motion
planning task immediately reduces to a discrete graph problem: After
labeling each edge of V (S) with its minimum distance to its two sites,
and adding s and t as new vertices to V (S), a breadth first search from s
can find, within O(n) time, a collision-free path to t in V (S) if one exists.
(Those edges whose clearance is less than the robot’s radius are ignored
during the search.)
If the target point, t, does not lie on any edge of V (S), we first determine
the line segment (t) whose Voronoi region contains t; to this end, pointlocation techniques as mentioned in Subsection 8.1.1 can be applied. Next,
we find the point z(t) on (t) that is closest to t; see Figure 8.18. If its
distance to t is less than the robot’s radius then the robot cannot be placed
at t and no motion from s to t exists. Otherwise, we consider the ray from
z(t) through t. It hits a point t on V (S), which serves as an intermediate
target point.
Similarly a point s can be defined if the original start point, s, does
not lie on an edge of V (S). Now the formerly infinite problem of finding a
collision-free path has become finite.
Theorem 8.6. The robot can move from s to t without collision iff the
following conditions hold : Its radius does not exceed one of the distances
d(z(t), t) and d(z(s), s), and there exists a collision-free motion from s to
t along edges of the Voronoi diagram V (S).
Proof. Suppose the conditions are fulfilled. Then the straight motion from
s to s does not cause a collision since the robot’s distance to its closest
obstacle is ever increasing. The same holds for t and t . Combining these
pieces with a safe motion from s to t yields the desired result. Conversely,
let us assume that a path π from s to r exists along which the robot can
move without hitting one of the line segments.
The mapping f : t → t can be extended not only to s but to all points
x of the scene (leaving fixed exactly the points of V (S)). One can show
that f is continuous. Consequently, path π is mapped by f onto a path π in V (S) that runs from s to t . For each point x on π, its corresponding
point x on π is even farther away from its closest obstacle.
This ‘retraction approach’ has first been taken by Ó’Dúnlaing and Yap
[569]. The Voronoi diagram of n line segments (Section 5.1) plus a structure
8.5. Motion planning
219
p
1
p
2
p
3
p
4
p
5
p
6
Figure 8.19. The polar Voronoi diagram for six point sites. Each region VR(pi ) is
bounded by two rays at pi : The negative horizontal ray for pi , and a ray defined by pi
and the unique site pj = pi with VR(pj ) pi . An exception is the region of the topmost
site, which is bounded by a single horizontal line.
suitable for point location in V (S) (Subsection 8.1.1) can be preprocessed
in time O(n log n). Afterwards, for each pair (s, t) of start and target points,
the above method can find a safe motion from s to t within time O(n), if
it exists, or report otherwise.
Several variants of the retraction approach are meaningful. For
example, convex distance functions (Section 7.2) can be used instead of
the Euclidean distance, to adapt to the shape of the robot.
We mention another variant, as it uses a nice type of Voronoi diagram,
introduced in Grima et al. [379, 380]. The polar angle of a point x with
respect to a point site p is the angle formed by the positive horizontal
ray through x, and the ray from x to p. We require x to have smaller
y-coordinate than p, so the polar angle is smaller than π. The polar Voronoi
diagram of a set of point sites allots to each site p the region of all points x
yielding the smallest polar angle; see Figure 8.19. Motion planning in a
specific direction now can be done by finding a chain of points such that
each of them can see its successor, using the polar Voronoi diagram in that
direction (not necessarily the horizontal one).
The polar Voronoi diagram is different from the visibility-constrained
Voronoi diagram in Section 5.4, where individual visibility angles are
attached to sites.
8.5.2. Translating polygonal robots
If approximating the robot’s shape by a disk is not accurate enough, a
compact convex set C can be used, with some fixed reference point in its
interior. As long as only translational motions are considered, the retraction
220
Chapter 8. Applications and Relatives
approach still works; only the Voronoi diagram of the obstacles should now
be based on the convex distance function defined by the convex set C that
results from reflecting C about the reference point; see Section 7.2.
If the scene consists of polygonal obstacles, the same situation occurs as
in the post office problem; see Subsection 8.1.1: Even though the number k of
obstacles may be small, the complexity of their Voronoi diagram increases in
proportion with their total number n of edges. Again, the piecewise-linear
approximation of the Voronoi diagram, the compact Voronoi diagram by
McAllister et al. [527] comes to the rescue; it is of complexity O(k) and can
be constructed within time O(k log n) for the Euclidean distance, and in
time O(k log n log m) for a distance function based on a convex m-gon. In
the latter setting, the diagram applies to solving the translational motion
planning problem of a convex robot with m edges.
Alternatively, planning translational motions for a convex robot R can
be reduced to motion planning for a point, using an observation by LozanoPérez and Wesley [512] described below.
Supposing that the origin o is a vertex of R, we replace each obstacle
Pi by the Minkowski difference
Pi R = {p − r | p ∈ Pi , r ∈ R}.
Geometrically, this amounts to moving the reflected image, R , of R around
Pi in such a way that o traces the boundary of Pi . In the resulting
configuration space, each point x not contained in any enlarged obstacle
Pi R corresponds to a collision-free placement of R in workspace. (This
space is also called the free space of the robot R.) As a consequence we
may, without loss of generality, assume that the robot is a point.
If the robot is also allowed rotational motions, Voronoi diagrams can
still be applied in some cases. Ó’Dúnlaing et al. [567, 568] have studied the
problem of moving a line segment amidst polygonal obstacles.
8.5.3. Clearance and path length
As mentioned earlier, a robot moving along the edges of the Voronoi
diagram maintains maximal clearance of the obstacles; however, the
resulting path may be longer than necessary. The shortest possible path,
on the other hand, has clearance zero because it does, in general, touch
obstacle boundaries. It can be obtained in time O(n2 ) by running Dijkstra’s
algorithm on the visibility graph of the n obstacle vertices; see Overmars
and Welzl [575] or Ghosh and Mount [361]. In this graph, two vertices
8.5. Motion planning
221
Figure 8.20. Visibility-Voronoi complex (dashed style) for the convex polygonal
obstacles s1 , . . . , s4 . Its straight edges form the visibility graph.
v, w are connected by an edge iff they ‘see’ each other, that is, if the line
segment vw does not intersect the interior of any obstacle polygon.
Wein et al. [700, 701] have presented different methods to obtain both,
sufficient clearance and short paths. In [700] a constant c ≥ 0 can be
specified, requiring that the robot must stay away a distance of at least
c from each obstacle at all times. Each obstacle si is enlarged by a belt of
width c, by forming the Minkowski sum si ⊕ C with a circle C of radius c,
that is, the c-offset of C. Consult Figure 8.20.
By construction, any shortest path in the visibility graph of the grown
obstacles is an optimum trajectory of the required clearance. Moreover, the
path is smooth, because obstacle vertices have been replaced with circular
arcs of radius c. But grown obstacles may overlap, thus closing a narrow
passage that existed before. In such a case, the user might rather accept a
smaller clearance than not find a collision-free path at all. To this end, the
Voronoi diagram of the original obstacles is added to all belt intersections.
Its edges may be assigned a weight higher than their Euclidean length, as
a penalty for their lack of clearance.
As the parameter c grows from 0 to ∞, the hybrid structure
just introduced (called the visibility-Voronoi complex ) changes from the
visibility graph of the original obstacles into their Voronoi diagram. After
222
Chapter 8. Applications and Relatives
O(n2 log n) preprocessing time, the optimum path for a given clearance c
can be computed in time O(n log n + k), where n is the total number of
obstacle vertices and k = O(n2 ) denotes the number of edges in the visibility
graph.
An alternative approach was taken in Wein et al. [701]. To combine
both length and clearance of a path π(t) into one measure of quality, they
introduced a weighted length
L(π) =
π
1
dt,
c(π(t))δ
where c(z) denotes the distance of point z to its nearest obstacle, and
δ > 0 is some weight. Surprisingly, optimal paths are smooth, and Voronoi
edges of polygonal obstacles are locally optimal ; but the sharp bends at
Voronoi vertices prevent the entire Voronoi diagram from being optimal.
In a discretization of the weighted length, one would introduce regions of
constant clearance. Where two regions of different clearance values share
an edge, Snell’s law of refraction applies to optimal paths.
2
For δ = 1 one can, in O( Λε2 n) time, construct a path whose weighted
length is by at most O(n) · ε larger than optimal. Again, n is the total
number of all obstacle vertices, and Λ equals the total weighted length of
a substructure of the Voronoi diagram.
8.5.4. Roadmaps and corridors
Instead of using the Voronoi diagram of the obstacles as a retract of the
free space where a robot may be placed, randomized roadmaps were studied,
e.g., in Overmars [574] and in Amato and Wu [56]. These structures can be
built incrementally by drawing points at random. If the new sample point
lies in free space, one tries to connect it by a straight edge to an existing
point of the roadmap. If this edge does not intersect any obstacle, it is
added to the structure, otherwise the edges (or the point) are discarded. In
this way a graph results that can be used for path planning.
Clearly, the sampling technique is critical. One would hope to reduce
the number of edges that must be discarded, and to increase the sampling
density near narrow passages. Yershova et al. [714] suggest to maintain,
for each vertex v of the structure already computed, a visible Voronoi
region containing all points of free space that are closest to, and visible
from, v. New sample points are chosen from a uniform distribution over
these regions. The appropriate structure reflecting this distance information
is the visibility-constrained Voronoi diagram in Section 5.4.
8.5. Motion planning
223
Figure 8.21. A roadmap built from sampling points of the obstacles’ medial axis, and
the medial axis segments connecting them (i). The corridor defined by two sample points
and its backbone path (ii). In the absence of further obstacles, the robot might follow
the solid path which results from a potential field approach. But the corridor is large
enough for the robot to locally avoid smaller fixed or mobile obstacles; see [359].
Geraerts and Overmars [359] suggest choosing sampling points from
the medial axis (see Chapter 5) of the large, static obstacles of a scene.
Figure 8.21 gives an illustration. They use them to build backbone paths
of corridors. A corridor is the union of all disks whose centers z lie on the
backbone and whose radii equal the clearance c(z) of z, up to a global
upper bound. Agents move in corridors, but need not follow fixed paths.
Instead, the corridor’s width provides some flexibility that can be used for
coordinating motions of groups, or for local obstacle avoidance. To this end,
potential-field methods can be employed. Wein et al. [701] proved that their
method for computing (near) optimal clearance-weighted paths mentioned
in Subsection 8.5.3 also applies to constructing optimal corridors.
There are several other applications of Voronoi diagrams in motion
planning. We refrain from more details here and refer the interested reader,
e.g., to the work by Canny [180, 181] and Choset and Burdick [220, 221].
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Chapter 9
MISCELLANEA
9.1. Voronoi diagram of changing sites
Constructing the Voronoi diagram of a fixed set S of sites is an algorithmic
problem which is inherently static. Several applications require data
structures which can represent proximity information that changes over
time, however. To name a few, one might need to preserve an optimal cluster
structure in a database for multivariate data that undergoes insertions and
deletions (Section 8.4), maintain clearance for a robot moving among nonstationary obstacles (Section 8.5), or simply wish to visualize the influence
a shopping market would exert on its competitors when being closed down
and re-opened at a promising location (Section 9.2).
Basically there are two — algorithmically quite different — possibilities
of altering the underlying set S of sites, which we will review in this section.
9.1.1. Dynamization
Discrete changes involve insertions and/or deletions of sites, and require
a dynamization of the data structure for Voronoi diagrams, like dynamic
search trees are used for storing and maintaining univariate (onedimensional) data [234, 538].
Usually, the sequence of insertions and deletions is not given in advance.
Updates have to be performed online, without information about future
requests. Note that deleting a site at location x and re-inserting it at
location y may simulate ‘jumping’ of a site from x to y.
We have seen ways of dynamizing the planar Voronoi diagram in
Sections 3.2 and 3.3 and also in Subsection 6.5.3 — based on the algorithmic
paradigms incremental insertion and divide & conquer — where the
construction history of the diagram, or of its dual Delaunay triangulation, is
stored for later use as a search structure. Dynamic geometric data structures
225
226
Chapter 9. Miscellanea
are well understood, and efficient solutions are available. In particular,
efficiency can be gained by using diverse randomization methods.
In our context, we refer to Devillers et al. [264], Schwarzkopf [624],
and the monograph by Mulmuley [556], for handling randomized update
sequences of sites in O(log n) expected time per operation. The two latter
references also treat convex hull maintenance in general dimensions, further
investigated in Clarkson et al. [228] and in Chan [190] — results of direct
relevance for dynamizing higher-dimensional power diagrams, and Voronoi
diagrams as a special case, by Theorem 6.2.
Also in [556], the problem of dynamic point-location in a planar Voronoi
diagram is discussed in detail. This gives a solution to the post-office
problem (Subsection 8.1.1) for a dynamically changing set S of sites. Among
several methods, the powerful technique of random sampling, introduced to
computational geometry by Clarkson [224], is applied to a triangulation of
the Voronoi regions radially from the sites. In a nutshell, this technique
randomly (and recursively) chooses a subset SR of S of suitable size,
constructs the desired data structure for SR , and bases the structure for S on
that substructure. The resulting hierarchy has a depth of O(log n) and a total
size of O(n), provided a constant fraction of sites is eliminated at each step.
Random sampling techniques lead to many efficient algorithms in
computational geometry but are usually difficult to analyze, and touch upon
complex concepts from the theory of ε-nets and discrepancy; see e.g. the
monographs by Matoušek [520] and Chazelle [195].
In our case, an update time of O(log n) and a point location time
of roughly O(log2 n) can be achieved for the post office problem, in the
expected sense.
A simpler but slightly less efficient approach, which also allows
for k-nearest neighbor searching, has been taken in Aurenhammer and
Schwarzkopf [102]. It is based on the history of randomized incrementally
constructing the order-k Voronoi diagram, and yields expected update
time O(k 2 log n + k log2 n) and expected query time O(k log2 n). The
space consumption is the same as for the diagram, O(k(n − k)); see
Subsection 6.5.3 for details.
More methods for dynamic k-nearest neighbor searching (and also for
dynamic point location in general ‘non-Voronoi’ planar subdivisions) can
be found in Mulmuley’s book [556], which offers a careful treatment of
geometric algorithms based on various randomization techniques.
9.1.2. Kinetic Voronoi diagrams
Continuous changes in S need to be modeled in various applications as
well. The sites now move steadily on predefined curves, their trajectories.
9.1. Voronoi diagram of changing sites
227
Such Voronoi diagrams of moving sites are so-called kinetic data structures,
a concept that has received considerable attention in the past fifteen
years; see e.g. Basch et al. [117, 118], Guibas [388], and Kirkpatrick
et al. [459].
In the kinetic Voronoi diagram, regions change shape continuously,
whereas the Delaunay triangulation DT(S) changes only in a discrete way
from time to time, when an edge flip (Section 3.2) has to restore local
Delaunayhood at the moment of cocircularity of four moving sites. For this
reason, implementations prefer to maintain the graph DT(S), and derive
the primal structure V (S) only when demanded.
The geometric aspect of kinetic Delaunay triangulations is almost
trivial. Delaunay edges admit local certificates of their validity, due to
their empty-circle property: By Thales’ theorem, an (interior) edge e in the
moving triangulation stays locally Delaunay (hence globally Delaunay by
Theorem 4.2), until the sum of the two angles opposite to e in the incident
triangles exceeds π. This can be calculated in O(1) time under reasonable
assumptions on the trajectories. A priority queue of ‘lifespans’ of Delaunay
edges with at most 3n − 6 entries can be kept, in order to predict the next
edge to be flipped, and to store the lifespan of the newly flipped-in edge, in
O(log n) time.
In its combinatorial aspects, the setting is surprisingly complex. Several
authors considered the kinetic Voronoi (or Delaunay) problem, even prior
to its first attack by computational geometers, in Guibas et al. [391].
How many Delaunay flips may occur in the worst case? In the paper
above, a nearly cubic upper bound of O(n2 λs (n)) is shown. Here λs (n)
is the maximum length of an (n, s)-Davenport–Schinzel sequence, and s
is a constant depending on the algebraic degree of the trajectories of the
point sites. The function λs (n) is superlinear in n for s ≥ 3, see Sharir and
Aggarwal [642], which includes all cases of interest here. Still, for constant
speed on straight lines, the bound above was improved to O(n3 ). A quick
lower bound — valid even for that simplest case — is Ω(n2 ), the number
of combinatorial changes of the convex hull of S; see Agarwal et al. [13]. In
comparison, the naive upper bound for Delaunay flips is O(n4 ).
Albers et al. [47] generalize the bound in [642] to higher dimensions,
and also consider the case where only k sites are moving and n − k stay
fixed, a scenario sensitive to the ‘fluctuation’ in S. See also Huttenlocher
et al. [414], who analyze the case where several groups of sites move rigidly
in the plane.
Very recent progress was made in Rubin [611], who proved what has
been conjectured for a long time (and what is true in the L1 -metric;
see Chew [214]): Only O(n2+ε ) flips can occur, under certain restrictions
on the cocircularities and collinearities that may arise from choosing the
228
Chapter 9. Miscellanea
sites’ trajectories and velocities. Still, the most basic case of constant
speed on straight lines is not yet covered, and remains a challenging open
question.
Agarwal et al. [12] proposed a way to relax the problem. They restrict
attention to a so-called stable subgraph of DT(S) which is less volatile. It
includes each Delaunay edge whose endpoints see the dual Voronoi edge
at a (necessarily equal) angle of at least α, for some appropriate value
of α. This graph need not contain any of the subgraphs of DT(S) discussed
in Section 8.2, but rather consists of certain edges of the shape Delaunay
tessellation (Subsection 7.2.2) for a regular polygonal shape. The stable
Delaunay subgraph undergoes only a near-quadratic number of changes,
each of cost O(1) in the implementation. The hidden constant depends on
parameter α.
We conclude this section with a few closely related results.
Fan et al. [334] consider kinetic variants of the Voronoi diagram for
graphs, which decomposes a given network into subparts owned by the
individual sites; see Subsection 9.4.1. Sites are moving continuously along
the network, and nearest-neighbor queries (i.e. the post office problem) and
k-nearest neighbor queries are to be responded to.
Of kinetic nature are also certain algorithms for constructing (static)
Voronoi diagrams. For instance, Gold et al. [366, 367] compute the Voronoi
diagram for line segments (Section 5.1) by growing points into line segments,
after having precomputed the classical Voronoi diagram for one segment
endpoint each.
Similarly, Kim and Kim [453] let sphere radii, i.e., additive site weights,
grow one by one, in order to obtain the Voronoi diagram of spheres in
3-space (Subsection 6.6.3).
Finally, Shewchuk [650] considers the problem of ‘repairing’ Delaunay
tessellations and regular complexes (Section 6.3), which arises in mesh
generation applications. He proves that, when such a simplicial complex
is distorted to non-Delaunay or non-regular, but only in a measurably
moderate way, then O(n) bistellar flips will restore the property.
9.2. Voronoi region placement
Let S denote a set of n points in the plane representing supermarkets. If
products are identically priced at each market, customers usually shop at
the supermarkets nearest to their residences. Thus, the Voronoi diagram
V (S) shows the market areas of the individual shops. (Were products
sold at different prices, we could model the situation by an additively or
multiplicatively weighted Voronoi diagram, as studied in Section 7.4.)
9.2. Voronoi region placement
229
If we assume that customers are uniformly distributed, a market’s
rate of sale is proportional to the area of its Voronoi region. Hence,
many important problems in economy translate into geometric questions
concerning Voronoi diagrams and related structures. The same is true for
diverse applications in the natural sciences, where Voronoi regions may
model, e.g., the extensions of habitats of biological species, or the structural
behavior of crystalline solids.
9.2.1. Maximizing a region
A typical facility location problem is how to place a new supermarket
where it wins over as many customers as possible from the competitors
that already exist. This task amounts to finding a position p for the new
market such that the area of its Voronoi region VR(p, S ∪ {p}) against its
competitors in S is maximized.
Two aspects of this problem statement need to be clarified. First,
if the new position p were chosen outside the convex hull of S then its
Voronoi region would be unbounded. There are several ways to deal with
this problem. One could require that p must be chosen ‘within city limits’,
that is, within some area F contained in the interior of the convex hull of S.
Or one could use a non-uniform customer distribution to the same extent.
Another option is to study the problem on a bounded surface, like, e.g., the
sphere or the torus; see Section 7.1.
Second, the Voronoi region of p is formally undefined if p coincides with
an existing location q in S. For our maximization problem, we could assume
that p moves towards q along a halfline ending in q. Then the bisector
B(p, q) of p and q converges to the line through q and orthogonal to . It cuts
the former Voronoi region VR(q, S) into two parts. The part intersected by
would become the Voronoi region VR(p, S ∪ {p}) of p. Even if it were not
allowed to place p exactly onto an existing point, a close approximation
would be possible.
Clearly, p can steal (close to) half of the area of VR(q, S ∪ {p}), by
choosing a suitable ‘line of attack’ against q, and even more than half if
VR(q, S ∪ {p}) is not point-symmetric about q. In total time O(n) one can
determine the largest region that p can obtain by attacking a point q ∈ S
in this way.
An even larger region might be secured by placing p at some point
not in S; see Figure 9.1. Let A(p) denote the area of VR(p, S ∪ {p}) as a
function of p. As long as p stays within the same cell of the arrangement
of all circumcircles of the Delaunay triangles in DT(S) (see Theorem 2.1),
the combinatorial structure of the Voronoi diagram does not change. In
Chapter 9. Miscellanea
230
Figure 9.1.
Placing a large Voronoi region within a bounding circle.
particular, the Voronoi neighbors of p, that is, the points pi ∈ S for which
VR(p, S ∪ {p}) and VR(pi , S ∪ {p}) share a Voronoi edge, are stable.
On such an arrangement cell, area A(p) can be expressed by the
Cartesian coordinates of the Voronoi vertices vi situated on the boundary
of VR(p, S ∪ {p}). If vi = (xi , yi ), where 0 ≤ i ≤ m in counterclockwise
order, then
A(p) =
m−1
1 (xi yi+1 − xi+1 yi )
2 i=0
holds, reading indices modulo m. The Voronoi vertices can be obtained
as intersections of bisector lines. As they depend on p = (px , py ), their
coordinates xi and yi are rational functions, whose numerators are
polynomials containing linear and quadratic terms in px and py , while the
denominators are linear in either variable; see Okabe et al. [571]. The area
function A(p) can be shown to be continuously differentiable for p ∈ S,
even when the Voronoi diagram undergoes combinatorial changes due to
the motion of p; see [571] and the references provided there. But despite
this favorable property, it is not known how to compute the exact maximum
of A(p).
Dehne et al. [249] proved the following partial result. Let N denote
the set of Voronoi neighbors of the new point p in S, and let P (N ) be the
9.2. Voronoi region placement
231
star-shaped polygon around p with vertex set N . Moreover, let LN be the
locus of all positions p that have N as their neighbor set.
Theorem 9.1. Suppose that the Voronoi neighbor set N of p is in convex
position. Then the area function A(p) attains at most one local maximum
on the polygon P (N ) ∩ LN .
Proof. (Sketch) It is sufficient to show that A(p) has at most one
maximum on each line, G. As P (N ) is convex, area A(p) can be expressed in
terms of P (N ) and the triangles formed by p and by its Voronoi neighbors
pi , rather than by the Voronoi vertices that vary with p. The sum of the
(signed) triangle areas has poles wherever line G crosses a supporting line
through neighboring points pi , pi+1 of P (N ), because here the Voronoi
vertex associated with pi , pi+1 and p is at infinity. By convexity of P (N ),
line G can pass only two supporting lines between their defining points pi
and pi+1 . Let l and r denote these crossing points. Since no other crossings
occur on G between l and r, one can show by studying second-order
derivatives, that A(p) attains at most one maximum in the interval [l, r].
An approximate solution to the Voronoi area maximization problem was
provided by Cheong et al. [207]. They assume that the existing locations
in S are situated in the unit square under torus topology, such that all
Voronoi regions are bounded.
Let b denote the maximum distance between a location pi ∈ S and
a point v on its region boundary in V (S). If we place the new market at
position v, its distance to any site in S must be at least b, because pi is a
nearest neighbor of v. This implies
max A(p) ≥ A(v) ≥
p
πb2
,
4
the latter inequality holding because VR(v, S ∪ {v}) contains an empty
circle of radius 2b centered at v. Moreover, each point of VR(p, S ∪ {p})
is at most a distance b away from its nearest neighbor in S, implying a
distance ≤ b from p. Each candidate region VR(p, S ∪ {p}) is, therefore, of
diameter ≤ 2b.
The main idea is to approximate the real area A(p) by the number of
vertices of an ε-by-ε grid Γε that are contained in VR(p, S ∪ {p}), times ε2 .
By the diameter bound, the approximation error cannot exceed 8 b ε.
Let popt be a point maximizing A(p), and let papp be a point whose
Voronoi region contains a maximum number µapp of grid points of Γε . Let
Chapter 9. Miscellanea
232
µopt denote the number of grid points in the region of popt . Then,
A(papp ) ≥ µapp ε2 − 8 b ε ≥ µopt ε2 − 8 b ε
4 πb2
≥ A(popt ) − 16 b ε ≥ A(popt ) − 16 b ε · 2 ·
πb
4
64 ε
≥ 1−
· A(popt ).
πb
Here the approximation bound has been used twice. The last line follows
from the above estimate for the region area. It remains to determine the
point papp .
Theorem 9.2. A point papp approximating the maximum Voronoi area to
a factor of 1 − δ can be found in time O( δn4 + n log n).
Proof. In time O(n log n) we construct the Voronoi diagram V (S) and
determine the value of b. For the sake of accounting, a grid Γb of square
cells of size b is superimposed to Γε . The complexity of Γb is O(n), since
each cell is within distance 2b of a site in S, by definition of b. If p is a
candidate point in some cell Q of Γb , its Voronoi region and its Voronoi
neighbors are contained in Q and its 24 neighboring cells, by the diameter
bound on VR(p, S ∪ {p}). Let SQ denote the set of points of S situated in
these 25 cells.
For a fixed cell Q of Γb and an arbitrary grid point g of Γε in Q, let
Wg = {p ∈ Q | g ∈ VR(p, S ∪ {p})}.
The set Wg is the largest disk centered at g that contains no interior points
of S, intersected with Q. For each of the N = b2 /ε2 grid points g in Q we
determine Wg by checking the distance from g to the elements of SQ . Next,
we construct the arrangement of all shapes Wg in time O(N 2 ), and find
a cell of this arrangement contained in a maximum number of these sets.
A point pQ of such a cell contains a maximum number of grid points in its
Voronoi region, over all points in Q. Point papp is obtained by maximizing
over all cells Q. The running time is O(n log n + nN + N 2 ), and for ε =
1
64 (π b δ) the claim follows.
9.2.2. Voronoi game
In Subsection 9.2.1 we have studied the situation where the area of one
Voronoi region was to be maximized. There the difficulty of an exact
solution lies with the analytical complexity of the area function. Now we
will consider problems involving several regions that have a game theoretic
flavor, too.
9.2. Voronoi region placement
233
Suppose two players, White and Black, can place n point sites each
in some bounded area. Their game proceeds in rounds. In each round,
first White and then Black are allowed to place a certain number of their
points, subject to specific rules. Points must not lie upon each other. When
all points have been positioned, the Voronoi diagram of the 2n points
is computed, and the total areas of all Voronoi regions owned by White
and Black, respectively. The player wins whose regions occupy the larger
total area. This example of competitive facility location is called a Voronoi
game.
If we assume for a moment that only the vertices of a very fine grid are
legal site locations, each player would have only a finite number of moves
when his turn comes. As in all finite zero-sum games of two players with
complete information, either White has a winning strategy, or Black does,
or both can force a draw. For general Voronoi games in two-dimensional
domains, it seems not to be known which case applies (a situation as for
chess). But there are interesting results for special cases.
Ahn et al. [23] have studied the one-dimensional case where players
compete for a circle or for a line segment of unit length. They proved the
following.
Theorem 9.3. On a circle or line segment, Black has a winning strategy
for the Voronoi game of n > 1 rounds. But against any strategy of Black,
White can secure an area of 12 − ε, for any given ε > 0.
Proof. (Sketch) In the analysis, a set of n equidistant ‘key’ locations plays
an important role. To win on a circle, Black plays onto empty key points
as long as possible. Then, he keeps splitting largest intervals that have two
white endpoints. With his last point, if only one such white interval, of
length , is left, Black plays into a bichromatic interval, less than n1 − away from its white endpoint.
ε
from the
To defend himself, White can place his points at distance 2n
1−ε
key points. In this way, each white interval has length ≥ n , and each
black interval is of length ≤ 1+ε
n . Since there are as many white as black
intervals, and because bichromatic intervals are fairly split among the two
.
players, Black’s excess is at most 2ε(n−1)
n
One should observe that Theorem 9.3 does not hold for one-round
games. On the circle, White can win by placing his points exactly onto
the n equidistant key points. Afterwards, Black can form at most n2 black intervals of length < n1 each. A similar approach works for the line
segment.
234
Chapter 9. Miscellanea
Cheong et al. [209] were the first to investigate the one-round Voronoi
game in dimension two and higher. They proved the following result which
implies that Black has a winning strategy on the square.
Theorem 9.4. For any set W of n ≥ n0 points in the unit square, there
exists a set B of n points not in W such that, in the diagram V (W ∪ B),
the Voronoi regions of the points in B have total area at least 12 + c. Here,
n0 and c > 0 are independent constants.
Proof. (Sketch) If there are ≤ n2 regions in V (W ) covering an area of
> 12 + c in total, Black can take over nearly all of their area by attacking
each w ∈ W by two of his own points from either side, in the way described
in Section 9.2. Otherwise, most regions in V (W ) have about the same size.
In this case, one can make use of the fact that even a point s chosen at
random has a Voronoi region VR(s, W ∪ {s}) of expected area 1+β
2n , for
some constant β > 0. This fact can be generalized to more than one point.
Since Black has a winning strategy for the one-round Voronoi game
on a square while White can always win on a line segment, the question
arises where the situation changes as the square is squeezed into a segment.
Fekete and Meijer [338] gave a surprising answer. Let ρ ≤ 1 denote the
aspect ratio of the rectangle in which the game is played (that is, the ratio
of the lengths of its shorter and longer edges).
Theorem 9.5. On a rectangle of aspect ratio ρ, Black has a√winning
strategy for the one-round game with n points if n ≥ 3 and ρ > n2 holds,
√
or if n = 2 and ρ > 23 . Otherwise, White has a winning strategy.
The proof of Theorem 9.5 is based on the following observations. If the
Voronoi diagram of the white points, V (W ), contains a region which is not
point-symmetric about its site, then Black can win by attacking the points
in W ; cf. Subsection 9.2. One can conclude that White’s only chance for
winning is to play his points in a regular grid. Afterwards, Black can win
iff he is able to place one point p that conquers an area strictly larger than
1
2n in V (W ∪ {p}). Clearly, this condition is necessary. It is also sufficient,
because after placing p, Black can still steal close to half of its former region
area from any point in w ∈ W , by placing a black point next to w on the
opposite side of p. Since the regions in V (W ) were all of area n1 , the claim
follows.
Fekete and Meijer [338] also considered the one-round Voronoi game
in a polygon with holes. They proved that Black’s optimization problem is
NP-hard once White has played his points.
9.2. Voronoi region placement
235
9.2.3. Hotelling game
In the Voronoi game, the position of a point site is fixed once the point has
been placed. A scenario allowing for relocation of sites in some bounded area
is known as the Hotelling game, named after the economist H. Hotelling.
Here each single point site pi in S is owned by a different player Pi ,
for 1 ≤ i ≤ n. In each round, each of the players P1 , P2 , . . . , Pn can, in
this order, move his site to a new location that maximizes the area of
its Voronoi region, while all other sites remain fixed. The question is if
an equilibrium will be reached where no player can gain by moving his
site. Also a characterization of all equilibrial configurations would be of
interest.
For the one-dimensional case, some of these questions can be answered.
On a line segment or on a circle, a point site p with two neighboring sites q, q has a Voronoi region of size d(q, q )/2, independent of its precise location
between q and q . Here d denotes the length of the (straight or circular)
segment connecting q and q that contains p. For the circle, this implies the
following.
Observation 9.1. Let S be a set of n points on the circle, labeled p0 , p1 ,
. . . , pn−1 in counterclockwise order. Then S is at equilibrium iff
d(pj , pj+1 ) ≤ d(pi−1 , pi ) + d(pi , pi+1 )
holds for all indices i and j modulo n.
Indeed, if this condition were violated, it would be profitable to relocate
pi to the circular segment from pj to pj+1 . On the line segment, the situation
is more complicated because, for the two outermost sites, it always pays
to attack their neighbors. Eaton and Lipsey [298] obtained the following
result.
Theorem 9.6. For n ∈ {2, 4, 5} point sites on a line segment, there are
unique equilibrial states. For n = 3, no equilibrium exists. For each n ≥ 6,
there is an infinite number of equilibrial configurations.
See Figure 9.2. More illustrations and many further results, also
of experimental nature, can be found in Okabe et al. [571]. Given the
difficulty of computing the maximum area attainable in a single move (see
Subsection 9.2.1), it may not seem surprising that a general solution to the
spatial Hotelling problem in dimensions two and higher has not yet been
discovered.
236
Chapter 9. Miscellanea
Figure 9.2. In the Hotelling game on a line segment, the equilibria for 2, 4, and 5 points
are unique. For six points, the distance between points 3 and 4 can range from 0 to 14 ;
the extremes are shown on the right-hand side. It is interesting to observe that points at
equilibrium may have Voronoi regions of different size. See [298] for details.
9.2.4. Separating regions
The problem of optimally placing Voronoi regions becomes easier in
its combinatorial and topological setting, rather than in its geometric
formulation where region areas etc. have to be considered.
Imagine an agent network represented by n black point sites in the
plane. The goal is to disconnect the network as much as possible, in the
sense that black Voronoi regions become non-adjacent when white point
sites are being placed. Another motivation for questions of this kind arises
in economics, where new shops are to be built so as to interfere with existing
(and competing) shops as much as possible.
Several variants of such separating region problems have been studied
in the literature, in their original form as well as in their equivalent dual
form: Given the set B of black sites, place a set W of white sites such
that, in the Delaunay triangulation DT(B ∪ W ), the number of black
connected components is large. Ahmed et al. [20] report that, with n white
sites, Black can be made less connected than White, i.e., the number of
black components in DT(B ∪ W ) is larger than the number of white ones.
sites, White can get all black sites as his neighbors,
Similarly, with only 2n+2
3
if n ≥ 12. The intention behind the last task is to try to entice customers
away from existing sites (shops) with better service.
Maximizing the number of black components, such that each of them
consists of just an isolated black region (see Figure 9.3) results in the
following stabbing problem for Delaunay triangulations: Find a preferably
small point set W that ‘stabs’ all the triangles’ circumcircles in the black
triangulation DT(B), that is, each such circumcircle has to enclose at least
one point in W . Aronov et al. [70] give an upper bound of 2n − 2 on
|W |, in their paper studying the concept of witness Delaunay graphs; see
Subsection 8.2.3. Ahmed et al. [20] offer an alternative, 5 |M |, where M is
9.2. Voronoi region placement
237
Figure 9.3. The presence of the white regions prevents the black regions from being
neighbors in this Voronoi diagram (from [70]).
a perfect matching in DT(B). It is known that the Delaunay triangulation
is guaranteed to contain some perfect matching; see Subsection 8.2.1.
The best upper bounds so far have been proven in Aichholzer et al. [41].
Their geometric constructions yield the following:
Lemma 9.1. |W | ≤ 3n
2 can be achieved, and can be improved to |W | ≤
if the point set B is in convex position.
5n
4
They also argue that n − 1 white points are always necessary to ‘block’
the black regions, i.e., to completely isolate them. For, if W blocks B,
then DT(B ∪ W ) does not contain any edge connecting two black points.
That is, B is a so-called independent vertex set in this triangulation. But
|+1
DT(B ∪ W ) has to contain some perfect matching M , having n+|W
2
components (one of the components might be a vertex rather than an edge).
Since at most one point per component in M can be black, it follows that
|+1
, and thus |W | ≥ n − 1.
n ≤ n+|W
2
A kind of reverse problem, namely, the problem of getting all the white
Voronoi regions connected by removing some black sites, has been studied
in de Berg et al. [241]. More precisely, given sets W and B of white and
black sites, respectively, how to find a minimum-sized subset B ⊂ B such
that the white regions form a single connected component in the Voronoi
diagram of W ∪ (B \ B )? If there are only two white sites, |W | = 2, then
Chapter 9. Miscellanea
238
an O(n log n) solution exists. The problem becomes NP-hard in its general
form, when |W | is not a constant.
A combinatorial rather than a geometric setting also underlies the
Voronoi game (cf. Subsection 9.2.2) on graphs, in Teramoto et al. [682].
Two players alternately occupy the n vertices of a given undirected and
unweighted graph G. Vertices are assigned to players according to the
nearest neighbor rule, where distances are here measured in terms of
numbers of edges (as opposed to geometric path lengths) on shortest paths
in G. The player who finally dominates the larger number of vertices is
the winner. When G is a k-ary tree then it is possible to formulate a
winning strategy, or to show the tieness of the game. For a short treatment
of Voronoi diagrams (and Delaunay structures) on graphs, see Section 9.4.
9.3. Zone diagrams and relatives
9.3.1. Zone diagram
As a fundamental property of Voronoi diagrams, they induce a partition of
their underlying space: Voronoi regions are (interior) disjoint by definition,
so the regions define a packing in space. On the other hand, for each point
there exists at least one closest site, which implies that the (closures of
the) regions yield a covering of the space. The concept of zone diagrams,
introduced in Asano et al. [76] and to be discussed now, lacks the covering
property, thus leaving a ‘neutral zone’ not being owned by any site.
Let S be a set of n point sites in the Euclidean plane R2 . The zone,
Z(p), of a site p ∈ S is defined to consist of all points in R2 closer to p than
to the zone of any other site in S. More formally, for a subset Z ⊂ R2 , let
dom(p, Z) = {x ∈ R2 | d(x, p) ≤ d(x, Z)},
where d(x, Z) = inf z∈Z d(x, z). Clearly, dom(p, Z) is a convex set that
necessarily contains p, as it can be expressed as an intersection of halfplanes
which contain p. The zones are now required to fulfill


Z(q), for all p ∈ S.
Z(p) = dom p,
q∈S\{p}
Observe that the description of this system of zones is implicit, as each zone
is defined in terms of the remaining ones. So, the issues of existence and
uniqueness of such a system arise.
The following equilibrium interpretation of a zone diagram gives some
intuition about this structure. Citing [76], “...the notion of the zone diagram
can be illustrated by a story about equilibrium in an age of wars. There are n
9.3. Zone diagrams and relatives
239
mutually hostile kingdoms. The ith kingdom has a castle at the site pi and a
territory Z(pi ) around it. The territories are separated by a no-man’s land,
the neutral zone. If the territory Z(pi ) is attacked from another kingdom,
an army departs from the castle pi to intercept the attack. The interception
succeeds if and only if the defending army arrives at the attacking point
on the boundary of Z(pi ) sooner than the enemy. However, the attacker
can secretly move his troops inside his territory, and the defense army can
start from its castle only when the attacker leaves his territory. The zone
diagram is an equilibrium configuration of the territories, such that every
kingdom can guard its territory and no kingdom can grow without risk of
invasion by other kingdoms.”
Let us mention some basic facts about the behavior of zone diagrams
first. Obviously, pi ∈ Z(pi ) holds, and Z(pi ) is a convex set that is contained
in pi ’s region VR(pi ) in the classical Voronoi diagram of S. See Figures 9.4
and 9.5, where the Voronoi diagram for five sites and the corresponding
zone diagram are shown. The distance of a zone boundary point x to
the respective site equals the distance of x to the closest different zone.
However, we face a peculiar behavior of zones, which is counter-intuitive for
Voronoi diagram-like structures (though not for straight skeletons, which
are not defined via distances; see Section 5.3): Zones can gain territory
when new sites are added. For example, placing a new site very close to an
existing one may make other neighboring zones expand. This phenomenon is
quite realistic in the above ‘kingdom’ interpretation: The two close sites, as
being ‘enemies’, weaken each other, and the surrounding zones get relatively
stronger. As another unusual property, all zones may be bounded sets, which
does not happen for ‘normal’ Voronoi diagrams.
In the simplest case, we have only two sites p and q, and the two
resulting zones are bordered by what is called the trisector curves C1 and
Figure 9.4.
Voronoi diagram.
Figure 9.5.
. . . and zone diagram [76].
240
Chapter 9. Miscellanea
C2 of p and q. These curves are defined such that every point on C1 has
equal distance to q and C2 , and every point on C2 has equal distance to p
and C1 . Asano et al. [77] prove that trisector curves exist and are unique
analytic curves. It is not known whether these curves are algebraic or even
expressible by elementary functions. Interestingly, the intersection of C1 (or
of C2 , its mirror image) with a given line parallel to the segment pq can be
computed, to any desired precision, in time polynomial in the number of
required digits.
In the case of more than two sites, things are even more complicated.
An intuitive argument for the existence and uniqueness of the zone diagram
could be based on the following growth model. From each site, a crystal
is growing, everywhere on its boundary and at unit speed. As soon as
the distance of a boundary point x of site p’s crystal K(p) to some other
crystal K(p ) becomes as large as d(x, p), the growth of K(p) stops at x.
The set of stopping points resulting from the overall growth process should
now delineate the zone diagram. (Note the difference between this growth
process and the process defining the crystal growth Voronoi diagram, in
Subsection 7.4.3.)
Theoretically easier to deal with, Asano et al. [77] prove existence
by iteratively approximating the zones, as follows. Consider an (ordered)
system A = (A1 , . . . , An ) of non-empty subsets of R2 , such that each Ai
covers exactly one site pi . Define


DOM(A) = (B1 , . . . , Bn ) with Bi = dom pi ,
Aj , for all i.
j=i
Then, the system A describes the zone diagram if A = DOM(A), that is,
if A is a fixed point of the operator DOM.
To approximate the zone diagram, we now start with the smallest
possible system, Z0 = ({p1 }, . . . , {pn }), which consists just of singleton
sets. When applying the operator once, we get
Z1 = DOM(Z0 ) = (VR(p1 ), . . . , VR(pn )),
the regions of the sites pi in the classical Voronoi diagram. In the next
system obtained, Z2 = DOM(Z1 ), each domain is bounded by curves
equidistant from a particular site pi and the union of the remaining
Voronoi regions, j=i VR(pj ). The domains in Z2 thus have piecewise
parabolic boundaries, and are clearly contained in the desired zones for
the sites. In general, applying the operator DOM iteratively, we get a
sequence Z0 , Z2 , Z4 , . . . of systems which approximates the zone diagram
from inside, and another sequence Z1 , Z3 , Z5 , . . . which approximates the
9.3. Zone diagrams and relatives
241
zone diagram from outside. Both sequences converge to the same limit, the
system Z = (Z(p1 ), . . . , Z(pn )) of unique zones for these sites.
This has been shown in [77], by applying Schauder’s fixed-point
theorem. They also outline an algorithm that constructs polygonal
approximations of the various obtained domains. No estimates on the
convergence rate of this algorithm are known, however. Alternative and
simpler proofs of existence are provided in Kopecká et al. [473] and in
Imai et al. [419]. The latter paper also establishes the existence of k-sector
curves for general k ≥ 2. In this context, we refer to an interesting paper
by Reem [603] that addresses several existence and stability questions for
general Voronoi diagrams.
The combinatorial size of the zone diagram is, of course, an interesting
quantity. By analogy with Voronoi diagrams, we can define a vertex of a
zone Z(pi ) as a point on the boundary of Z(pi ) with at least two distinct
closest points in j=i Z(pj ). Edges are boundary pieces that connect two
consecutive vertices of a zone. An edge e of Z(pi ) is always part of the
bisector of pi and some fixed zone Z(pj ). The total number of edges and
vertices can be shown to be O(n). Still, this seemingly harmless complexity
is misleading, as e might have a complicated structure stemming from
boundary parts of Z(pj ) that contain many vertices.
Generalizations of the zone diagram have been considered in Kawamura
et al. [448]. They prove uniqueness for disjoint and compact sites of general
shape in d-space, and also in certain normed spaces where unit balls
are smooth and strictly convex. (Compare Section 7.2 on convex distance
functions.) On the other hand, uniqueness is already lost, for example, for
the L1 -norm in the plane, because its unit circles are not smooth.
9.3.2. Territory diagram
A modification of different flavor, called territory diagrams, is studied in
de Biasi et al. [258]. Compared to the definition of a zone diagram, set
equality is now replaced by set inclusion. More specifically, given point
sites p1 , . . . , pn , any system of territories T1 , . . . , Tn in R2 with pi ∈ Ti is
called a territory diagram (for these sites) if it fulfills the condition

T (pi ) ⊆ dom pi ,

T (pj ),
for i = 1, . . . , n.
j=i
Territory diagrams trivially arise whenever T (pi ) ⊆ Z(pi ) holds, with Z(pi )
being the zone of pi . In particular, any zone diagram is a territory diagram.
Also, in the sequence of systems Z0 , Z2 , Z4 , . . . used above to (convexly)
242
Chapter 9. Miscellanea
approximate zones from inside, each system Zi constitutes a trivial instance.
Valid territories need not be convex, however. In fact, non-trivial territory
diagrams exist, as a simple example with two sites shows, where T (p1 ) ⊂
Z(p1 ) but T (p2 ) is not contained in Z(p2 ).
Territory diagrams are ‘mollified version’ of zone diagrams, best
explained in the ‘kingdom’ interpretation given before. Suppose that all
the kings in the country have come to an agreement that, after one of them
claims new territory, this will not be contested by others — if it does not
lie closer to any’s territory location x ∈ T (pi ) as is the distance from x to
pi (since that would allow the respective king to successfully defend x from
his castle, pi ). This regulation enables a particular king to conquer parts of
the country larger than delineated by its zone Z(pi ) in the zone diagram,
by claiming territory sooner than his antagonists.
A territory diagram is called maximal if none of its territories can be
expanded without violating the definition. It is clear that zone diagrams
are maximal territory diagrams. The converse, however, is not true, and
territories in a maximal territory diagram may still be nonconvex. The
existence of maximal territory diagrams can be established via Zorn’s
lemma on partial orders, which leads to a somewhat simpler existence proof
for zone diagrams.
9.3.3. Root finding diagram
A structure apparently similar to planar Euclidean zone diagrams arises
in Kalantari [438], in the context of polynomial root finding. Let us view a
point pj = (xj , yj ) in R2 as a complex number
√
zj = xj + i · yj , for i = −1.
We can now describe a set of n point sites in the plane as the roots of a
degree-n polynomial Q(z) in the complex numbers,
n
(z − zj ).
Q(z) =
j=1
Conversely, given a polynomial Q(z), root finding in the complex plane can
be done by starting with some initial value z0 and applying one of the known
iterative methods. For suitable choices of z0 , convergence to some root zj of
Q(z) will be achieved. That is, the corresponding point p0 will be allotted
to the site pj . The set of all points in R2 whose orbit (under the iterations
of a particular root finding method) converges to pj is called the basin of
attraction, A(pj ), of pj . Depending on the method used, A(pj ) is a more or
9.3. Zone diagrams and relatives
243
less accurate approximation of the Voronoi region VR(pj ). Unfortunately,
A(pj ) typically has a complicated structure, being disconnected, and even
of fractal nature at its boundary. The connected component of A(pj ) that
contains pj is called the immediate basin of attraction of pj .
It is challenging to design root finding methods whose immediate basins
of attraction resemble the shapes of Voronoi regions as much as possible.
This goal is pursued in [438], by constructing a sequence Bm (z), m ≥ 2,
of iteration functions whose first two members, B2 (z) and B3 (z), are well
known as Newton’s method and Halley’s method, respectively. For m → ∞,
the basins of attraction of Bm (z) uniformly approximate the Voronoi
diagram of the roots from inside, to within any prescribed tolerance. That
is, the ‘chaotic parts’ will lie in a controllable neighborhood of the Voronoi
edges. On the other hand, it is shown that no rational iteration function is
capable of achieving the same goal.
Using the sequence Bm (z), one can define certain ‘safe’ convex
subregions of the immediate basins of attraction, ordered by inclusion as m
grows and the uncovered parts of the plane shrink. These regions exhibit a
shape very similar to those in the zone diagram of the sites; see Figure 9.6.
Figure 9.6.
Layering of convex subregions of basins of attraction (from [438]).
244
Chapter 9. Miscellanea
Apart from the visual similarity, however, a formal link between the two
concepts — basins of attraction and zone diagrams — is still missing.
Each complex polynomial with n single roots will draw its individual
layering of safe subregions, a fact used in what is called polynomiography in
Kalantari [437]. This field of study is based on the computer visualization
of the process of solving polynomial equations, and has applications in art,
education, and science.
We remark at this place that Voronoi diagrams did not go unnoticed
in the world of art and design. For example, Kaplan [442] creates attractive
ornamental designs using Voronoi regions, Dobashi et al. [282] generate
mosaic images for a non-photorealistic rendering method that show an
artistic effect, and Vanhoutte explores the expressional power of fractal
Voronoi diagrams. An internet search for Voronoi art will reveal numerous
pleasing and colorful demonstrations of Voronoi diagrams as an artistic
tool.
9.3.4. Centroidal Voronoi diagram
In the iterative process for approximating zone diagrams, described in
Subsection 9.3.1, the sites remain fixed whereas the regions converge to
the desired zones. There is quite a different type of Voronoi diagram where,
in contrast to zone diagrams, the sites are allowed to move in the iteration
process until they reach ‘stable’ positions.
A so-called centroidal Voronoi diagram is a (Euclidean) Voronoi
diagram for point sites in a given convex domain Ω ⊂ Rd , such that
each site pi represents the centroid (center of mass) of its Voronoi region.
(Usually, the region centroid can be located quite far away from the defining
site.) Although probably not related to zone diagram structures, we briefly
discuss this interesting special instance of classical Voronoi diagrams at this
place, as it can be seen as the fixed point of a certain operator on Voronoi
diagrams as well.
Sometimes this concept has been also called self-centered Voronoi
diagram in the literature. This should not be confused with the notions
of self-centered tetrahedral (or simplicial) complexes in Section 6.1, and of
self-centered power diagrams in Section 10.5.
Centroidal Voronoi diagrams are usually defined for a given density
function ρ on Ω, that is, weighted centroids of the regions are considered.
The main question that arises is to find sets of sites, of given cardinality k,
such that their Voronoi diagram is centroidal with respect to ρ. Figures
9.7–9.9 (from [435]) depict three planar examples, where different density
functions are taken in the square [−1, 1]2 .
9.3. Zone diagrams and relatives
Figure
9.7. Centroidal
Voronoi
diagram,
for
ρ(x, y) = 1 (area).
Figure
9.8. Centroidal
Voronoi
diagram,
for
density function ρ(x, y) =
2
2
e−10(x +y ) .
245
Figure
9.9. Centroidal
Voronoi
diagram,
for
2
2
1
ρ(x, y) = e−20(x +y ) + 20
2
2
sin (πx) sin (πy).
Centroidal Voronoi diagrams do always exist, for any number k of sites.
However, they are ambiguous structures, even in their simplest form where
ρ(x) = 1 for all x ∈ Ω (uniform distribution), as a simple example with two
sites in R2 shows.
In their survey paper, Du et al. [289] discuss various applications of
this appealing diagram in diverse areas. Among others, they include data
compression in image processing (where ρ represents the color density),
estimation of data (for example in meteorology, where the sites represent
optimal locations for precipitation measurement), facility location (sites
indicate the optimal placement of resources), and territorial behavior in
biology (where synchronous settlement of animals tends to take place at
the sites of a centroidal Voronoi diagram).
An iterative construction algorithm, known as Lloyd’s method [511]
in the electrical engineering literature, alternates between constructing
Voronoi regions and (weighted) centroids. Let S0 be a set of k point sites
in Rd , possibly initially generated by a Monte Carlo method. For i ≥ 0,
we write Si+1 = OP(Si ), where the operator OP constructs the Voronoi
diagram of Si and then assigns to Si+1 the centroids of its regions. Clearly,
any fixed point of the operator OP corresponds to a centroidal Voronoi
diagram for k sites.
Several variants of Lloyd’s method, as well as other iteration schemes for
approximating centroidal Voronoi diagrams, are described in Du et al. [289]
and Du and Wang [290]. Computation costs are usually high, as they
may involve the construction of higher-dimensional Voronoi regions and
the evaluation of multi-dimensional integrals.
Alternative construction methods that avoid these drawbacks are given,
e.g., in Ju et al. [435]. Their probabilistic method, which is also suited to
parallelization, obtains accurate and efficient approximations of centroidal
246
Chapter 9. Miscellanea
Voronoi diagrams without the need of explicitly constructing Voronoi
regions.
Let us finally mention a related concept we have already encountered
earlier. When the underlying convex domain Ω is replaced by a finite
point set X, then the construction of a ‘centroidal Voronoi diagram’ for X
amounts to finding a k-centroid clustering of X; see Subsection 8.4.1. This
popular type of geometric clustering is also called k-means clustering in the
literature.
9.4. Proximity structures on graphs
In certain applications, it is not appropriate to model distances between
locations in the entire plane, but rather on a given connectivity structure.
In fact, this is a standard framework in the area of facility location, where
clients and suppliers lie on a connectivity network, usually a planar graph or
even a tree. The network may have to be specified as an abstract weighted
graph (where the triangle inequality does not hold, in general), or by some
geometric graph, usually with straight edges in the plane.
9.4.1. Voronoi diagrams on graphs
Graph Voronoi diagrams have first been considered in Mehlhorn [532], for
finding approximations of minimal Steiner trees in graphs. The underlying
model is an edge-weighted graph G = (V, E), where each edge has assigned
a positive weight (or cost). The distance, d(a, b), between two vertices a
and b is defined as the sum of edge weights on a shortest path in G between
a and b. For geometric graphs, weight can be just the Euclidean length.
Distances between any two points placed on edges of G, not just between
vertices, are meaningful then.
Given a subset S = {s1 , . . . , sn } ⊂ V of distinguished vertices (the
sites), the Voronoi diagram of G and S is a partition of V into subsets
Vi = {a ∈ V | d(a, si ) < d(a, sj ), j = i}.
(We ignore the degenerate case where equality of path lengths does occur,
which can be handled consistently.) Depending on the application, we might
want to store some more information on these ‘Voronoi regions’ Vi , like the
distance from each vertex a ∈ Vi to its defining site si , or the corresponding
set of shortest paths.
Some properties of graph Voronoi diagram are discussed in the book by
Okabe et al. [571], who call it the network Voronoi diagram. For example,
and unlike the situation in the Euclidean plane, regions lack a convexity
9.4. Proximity structures on graphs
247
property. Following the standard definition of graph convexity, a vertex
set U ⊂ V is called convex (with respect to G) if, for any a, b ∈ U , also all
the vertices on shortest paths between a and b are in U . In fact, the sets Vi
are not convex in general, but rather are star-shaped, in the sense that Vi
contains all vertices on shortest paths from a ∈ Vi to si .
Using an extension of Dijkstra’s shortest path algorithm, Mehlhorn
[532] computes the graph Voronoi diagram in time O(e + v log v), when G
has e edges and v vertices. Erwig [329] improves the running time slightly,
to O(e + (v − n) log(v − n)), where n denotes the number of sites in S,
and extends the algorithm to work on directed graphs. Note that, in the
directed setting, unreachable vertices might exist, which thus do not belong
to any Voronoi region; see Figure 9.10. Also, we have to distinguish between
the outward and inward graph Voronoi diagram, respectively, depending on
whether we want to consider path lengths from sites to vertices, or from
vertices to sites. Parallel Dijkstra is used in the construction algorithms,
based on sophisticated dynamic data structures like Fibonacci heaps. A
lower bound for computing the graph Voronoi diagram is Ω(max{v, (v − n)
log(v − n)}).
Hurtado et al. [412] consider Voronoi diagrams on geometric graphs,
mostly trees. Euclidean shortest path lengths are taken as distances,
but weighted multiplicatively, modeling the attractiveness of a site (see
Section 7.4). Sites may be placed anywhere on edges. They also discuss
the farthest-site variant, where vertices are allotted to sites farthest away
(rather than closest), and colored versions thereof.
4
1
4
1
1
1
2
1
s1
1
1
1
2
1
1
s1
1
1
1
2
1
3
1
2
s2
1
1
3
s2
1
Figure 9.10. Outward and inward Voronoi diagrams on a weighted directed graph, with
two graph vertices s1 and s2 as sites.
248
Chapter 9. Miscellanea
One motivation for studying graph Voronoi diagrams stems from
handling typical queries in databases for networks; see e.g. Erwig and
Güting [330]. For example, the graph G may model a transportation
network, and the set S of sites corresponds to vertices of G that represent
post offices, supermarkets, or fire stations. The graph versions of various
distance problems, like the post office problem, nearest neighbors and closest
pair, or ‘largest empty circle’ (Section 8.1) can now be solved efficiently.
In particular, the last problem is relevant for facility location and leads
to the notion of anti-center of a graph G with respect to S, which is
the subset of vertices farthest away from any site in S. Such vertices are
potential candidates for building properly located service plants. Given
the graph Voronoi diagram, the anti-center can be computed in O(v)
time. Note that the center of G, that is, the subset of vertices with the
smallest maximum distance to any vertex of G, corresponds to the ‘smallest
enclosing circle’ for G.
As one more application, we mention secure path planning (cf.
Section 8.5) in graphs, where a path from a source to a destination is to be
found that keeps clear as much as possible from a set S of ‘dangerous’ sites.
9.4.2. Delaunay structures for graphs
Consider a geometric straight-line graph G with n vertices in the plane, not
necessarily planar. We now measure distances as the number of edges along
shortest paths in G (not as the weighted path length, as done for the graph
Voronoi diagram in Subsection 9.4.1). The k-neighborhood of a vertex p is
the subset of G’s vertices at distance at most k from p, that is, the set of
vertices which can be reached from p with k ‘hops’. The k-locally Delaunay
graph for G, denoted by DT(G, k), includes every edge of G that can be
separated by some circle from the k-neighborhoods of its two endpoints.
See Figure 9.11 for an illustration.
The similarity of this concept to witness Delaunay graphs defined
in Subsection 8.3.2 should be noted. Witness Delaunay graphs are more
general, in the sense that the witness point set need not come from some
graph neighborhood; on the other hand, they are more restrictive, in the
sense that the underlying graph G is just the complete geometric graph.
k-locally Delaunay graphs have been introduced in Li et al. [505], for
constructing distributed topology control protocols for network routing in
mobile networks. As an underlying graph G, they assume the unit-disk graph
on a finite point set S in the plane, that is, the graph connecting any two
points in S at distance less than 2 (see also Section 10.5). This models
the situation where all wireless nodes have the same transmission range.
9.4. Proximity structures on graphs
249
e
Figure 9.11. Edge e is included in DT(G, 1), because e can be separated by some circle
from its direct neighboring vertices. No circle exists, however, that can separate e from
vertices at two hops away; therefore, e is not part of DT(G, 2).
By removing this restriction, and thus allowing individual transmission
radii, Kapoor and Li [444] study the structure of DT(G, k) for general
geometric graphs G.
The size of DT(G, k) is an interesting quantity, because dense graphs G,
though providing many edges as possible candidates for inclusion, also have
rich vertex neighborhoods which cause exclusion of edges. Obviously, the
graph DT(G, k + 1) is a subgraph of DT(G, k), that is, k-locally Delaunay
graphs get sparser with increasing k. Note the difference to higher-order
Delaunay graphs in Section 8.3.1, which show an opposite behavior.
If G is connected and k is sufficiently large, DT(G, k) becomes a
subgraph of the (classical) Delaunay triangulation DT(S) for the vertex
set S of G. In the extreme case where G is the complete geometric graph Kn
on S, we have DT(Kn , 1) = DT(Kn , k) = DT(S), for all k.
As the number of edges in DT(G, k) usually happens to decrease rapidly
with k, only small values of k are interesting. Basically, only the cases k = 1
and 2 are relevant in most applications. For unit-disk graphs, DT(G, 2) is
already a planar graph, and DT(G, 1) can be decomposed into two planar
graphs; see [505]. Bounds that are valid for arbitrary geometric graphs G
are given in Kapoor and Li [444], based on small forbidden subgraphs, as
we briefly describe below.
8
For example, they show that DT(G, 1) has O(n 5 ) edges, by applying
a result by Pinchasi and Radoičić [592], that any geometric graph with no
8
self-crossing cycle of length 4 has a size of O(n 5 ). To see that DT(G, 1)
is indeed such a graph, consider some self-crossing cycle, C4 . All what C4
contributes to DT(G, 1) must be a subgraph of DT(S)∩G, because for each
edge e of C4 , the remaining two vertices are reachable from at least one of e’s
250
Chapter 9. Miscellanea
endpoints within one hop. So, as DT(S) does not have edge crossings, C4
can have neither.
DT(G, 2) has already only O(n log n) edges, because this graph is free
of self-crossing paths of length 3. Let pqrs be such a path. Then, either
any circle through p and q contains r or s, or any circle through r and s
contains p or q. Therefore, not both of the edges pq and rs can belong to
DT(G, 2), which excludes the path pqrs from appearing in DT(G, 2). Pach
et al. [576] showed that the absence of such self-crossing paths restricts the
number of edges to O(n log n).
Pinchasi and Smorodinsky [593] improved the two upper bounds above,
using different methods, and also gave a lower bound. We summarize their
results.
3
Theorem 9.7. DT(G, 1) has O(n 2 ) edges, and there are examples where
4
Ω(n 3 ) edges arise. Furthermore, DT(G, 2) has at most 32n edges.
The graph DT(G, 1) has a surprising application, concerning
arrangements of so-called pseudo-lines and pseudo-parabolas; see Tamaki
and Tokuyama [674]. These are dissections of the plane, induced by open
curves which pairwise cross at most once (or at most twice, for pseudoparabolas). In fact, the number of curve segments that n pseudo-parabolas
need to be cut into, in order to obtain a pseudo-line arrangement, is
bounded by the number of edges in DT(G, 1).
Naturally, k-locally graphs for G can be based on subgraph concepts of
the Delaunay triangulation (Section 8.2). For example, the k-locally Gabriel
graph for G consists of all edges of G whose diametrical circle separates the
respective edge from the k-neighborhoods of its two endpoints. This graph
is sparser than DT(G, k), and has been preferably used in wireless network
routing applications [505, 444]. Gabriel graphs are also easier to deal with
algorithmically, because separating circles are fixed now to be diametrical
circles.
Chapter 10
ALTERNATIVE SOLUTIONS IN Rd
10.1. Exponential lower size bound
In the preceding chapters we have repeatedly stated that high-dimensional
Voronoi diagrams suffer from an exponential worst-case complexity; a
precise formula based on results by Seidel [625] is given in Theorem 6.1.
To give a quick argument for this fact, let us consider a set S of n
points on the moment curve M in d-space Rd , which is parametrized by
m(t) = (t, t2 , . . . , td ).
Consider a subset F of size d of S with the following evenness property:
Between any two points of S\F there is an even number of points of F
d
on M . Such subsets F of S can be chosen in Θ(n 2 ) many different ways;
see Matoušek [520] for a precise analysis. Let HF denote a hyperplane in
Rd containing F ; see Figure 10.1.
Hyperplane HF equals the set of all points (x1 , . . . , xd ) satisfying the
equation
a1 x1 + a2 x2 + · · · + ad xd = 0
with certain real coefficients a1 , . . . , ad . For each point m(t) of M that lies
on HF , the parameter t is a zero of the polynomial
a 1 t + a 2 t2 + · · · + a d td .
Since a polynomial of degree d has at most d roots, we obtain M ∩ HF = F .
Moreover, all points in S\F must be situated on the same side of HF , by
the evenness property. Hence, HF is a supporting hyperplane of the convex
hull of S (which is a so-called neighborly polytope [384] in this case), and
the points in F form one of its (d − 1)-dimensional facets.
For each such subset F , there is an (unbounded) Voronoi edge of V (S).
Namely, this edge is the union of the centers of all d-spheres that pass
251
Chapter 10. Alternative Solutions in Rd
252
M
HF
Figure 10.1.
Points in F are drawn in black, points of S\F in gray.
through the points of F but do not contain any point of S\F ; cf. the
proof of Lemma 2.2. Thus, V (S) has an exponential number of unbounded
edges.
d
A worst-case complexity of Θ(n 2 ) renders Voronoi diagrams useless
for computing, for instance, all nearest neighbors within a set of n points,
which could trivially be done in Θ(n2 ) time. Trusting that the given input
points will not lead to the worst case, one could apply an output-sensitive
method for computing the Voronoi diagram. Miller and Sheehy [543]
recently presented an algorithm that runs in time O(f log n log ∆), where
f is the total number of faces in the resulting Voronoi diagram V (S), and
∆ denotes the so-called spread of S, i.e., the ratio of the largest and the
smallest pairwise distances in S. This method first constructs the Voronoi
diagram of a superset of S for which a small complexity can be guaranteed.
Then the extraneous points are carefully removed.
The notion of spread of a point set leads to an interesting view of various
geometric problems. We refer to Erickson [327, 328] and references given
there. For instance, the size of the Delaunay triangulation DT(S) in R3
is bounded by O(∆3 ) if S has spread ∆, a quantity independent from the
size n of S; see [328].
In the subsequent sections, we will mention some alternative techniques
for solving distance problems for point sets in high dimensions.
10.2. Embedding into low-dimensional space
A natural question is if high dimensions can be avoided altogether
by mapping the given point set to some lower-dimensional space, and
solving all distance problems there. Such dimension reduction techniques
would work perfectly well if the pairwise distances among the points
could be exactly preserved under such mapping. The example of the
10.2. Embedding into low-dimensional space
253
regular tetrahedron shows that this is too much to hope for: Its vertices
have pairwise distance 1 in R3 , but no four points in R2 have this
property.
However, the above goal can be achieved with a small distortion, as
a famous result by Johnson and Lindenstrauss [432] shows. It claims the
existence of a global constant C such that the following holds. As usual,
d denotes the Euclidean distance.
Theorem 10.1. For each set S of n points p1 , . . . , pn in Rn , and for each
n
ε ∈ (0, 12 ), there exists a function f : Rn −→ Rk , where k = C · log
ε2 , such
that
(1 − ε) · d(pi , pj ) ≤ d(f (pi ), f (pj )) ≤ (1 + ε) · d(pi , pj )
holds for all 1 ≤ i, j ≤ n.
In other words, any n-point set in Rn can be mapped to RO(log n) such
that the interpoint distances change by a factor of at most 1 ± ε.
Proof of Theorem 10.1. (Sketch) In order to find low-distortional
mappings, Johnson and Lindenstrauss [432] considered the function
g : S n−1 −→ R
(x1 , . . . , xn ) −→ x1 2 + · · · + xk 2 ,
which, for each point on the sphere S n−1 in Rn , computes the Euclidean
length of its projection onto the first k coordinates. Let m = nk . One can
show that m is the median of g, which means that the set A of all a ∈ S n−1
for which g(a) ≤ m holds just covers half the surface area of S n−1 . Now a
surprising property of high-dimensional spheres applies: If a ‘hemisphere’
set like A is extended by a belt of width only t > 0, the resulting set
At = {q ∈ S n−1 | ∃ a ∈ A : d(q, a) ≤ t}
2
covers all of S n−1 , except for an O(exp(− t 4n ))th part. Therefore, a random
point q of S n−1 lies, with high probability, within distance t of some point a
satisfying g(a) ≤ m. Since the function g is 1-Lipschitz, g(q) ≤ g(a) +
d(q, a) ≤ m + t follows and, by a symmetric argument,
|g(q) − m| ≤ t.
In other words, function g is sharply concentrated around m. Instead of
projecting a random point onto a fixed k-dimensional subspace (given by
the first k coordinates), one can fix the point and choose a random linear
254
Chapter 10. Alternative Solutions in Rd
subspace H of dimension k in Rn , with orthogonal projection f onto H.
p1 −p2
∈ S n−1 and t = 3ε m. By
For two fixed points p1 , p2 in Rn , let q = d(p
1 ,p2 )
linearity of projection f , the previous inequality implies that
(1 − 3ε) · m · d(p1 , p2 ) ≤ d(f (p1 ), f (p2 )) ≤ (1 + 3ε) · m · d(p1 , p2 )
holds with probability at least 1 − n12 for dimension k = 300 e12 log n.
So, if we are given n points p1 , . . . , pn in Rn , the probability
that one
of the pairs pi , pj violates the previous estimates is less than n2 · n12 < 1.
Thus, the projection f onto a random subspace H is an embedding of low
distortion, with a positive probability. A complete proof can be found in
Matoušek [520].
For algorithmic applications, one would repeatedly generate a random
coordinate matrix of a mapping f , and test whether it is of low distortion.
In order to do this efficiently, the matrix should have a simple structure
and be sparse, that is, contain as many zero entries as possible. On the
other hand, the sharp concentration property must be preserved. Recent
work by Ailon and Chazelle [45] and by Matoušek [521] shows that these
goals can be achieved to some extent. Instead of a projection mapping,
they use a product M · H · D of three matrices, where, after scaling, D
is a diagonal matrix with independent random entries in {±1}, H is an
isometry given by a Walsh matrix with {±1} entries, and M is a sparse
k×n matrix each of whose entries equals zero with probability 1−q, and ±1
with probability q2 each, independently. The result of applying the product
M ·H ·D to some vector can be computed in time O(n log n), by fast Fourier
transform.
More information on metric space embeddings can be found, e.g., in
Indyk and Matoušek [425].
10.3. Well-separated pair decomposition
In this section, we discuss a structure that solves some distance problems in
high dimensions very efficiently; it has other interesting applications, too.
Throughout this section, S is a set of n points in Rd , and s > 1 denotes the
so-called separation constant. Both s and the dimension d are considered
fixed, while the number of points, n, is variable. Given a finite point set A,
we denote by R(A) the bounding box of A, that is, the smallest axis-parallel
hyperrectangle in Rd containing A.
(A hyperrectangle is the Cartesian product of d mutually orthogonal
line segments in Rd . If the lengths of all segments are the same, a
d-dimensional hypercube is obtained.)
10.3. Well-separated pair decomposition
Figure 10.2.
Sets A and B are well-separated with respect to s =
255
5
.
2
The results in this section are based on work by Callahan and
Kosaraju [177].
Definition 10.1. Two point sets A and B in Rd are said to be wellseparated (with respect to the separation constant s) if there exist two balls
CA and CB of radius r, each containing the bounding box of its respective
set, and being at least a distance of s · r apart.
A two-dimensional example is shown in Figure 10.2. It illustrates a
simple, but important consequence of Definition 10.1. Namely, points within
the same set of a well-separated pair are close, while points on opposite sides
are far, and have almost the same distance between them.
Lemma 10.1. Let A, B be well-separated with respect to s. Let a, a ∈ A
and b, b ∈ B. Then,
(1) d(a, a ) < 2 r ≤ 2 d(CAs,CB ) ≤ 2s d(a, b) and
(2) d(a , b ) ≤ d(a , a)+d(a, b)+d(b, b ) ≤ ( 2s +1+ 2s )·d(a, b) = (1+ 4s )·d(a, b).
The next definition is of central importance.
Definition 10.2. A sequence (A1 , B1 ), (A2 , B2 ), . . . , (Am , Bm ), where Ai ,
Bj ⊂ S, is called a well-separated pair decomposition of S with respect
to s if
(1) each pair (Ai , Bi ) is well-separated with respect to s, and
(2) for any two points p = q of S there is exactly one index i, 1 ≤ i ≤ m,
such that p ∈ Ai and q ∈ Bi hold, or vice versa.
The following lemma gives a first application of well-separated pair
decompositions (WSPDs, for short).
Lemma 10.2. Consider a well-separated pair decomposition of S with
respect to s ≥ 2, consisting of m pairs of sets. Then the closest pair of
points in S can be found in time O(m).
256
Chapter 10. Alternative Solutions in Rd
Proof. Let a, b be a closest pair in S. By Property 2 of Definition 10.2,
there exists a pair (Ai , Bi ) in the WSPD such that p ∈ Ai and q ∈ Bi . If there
were another point p of S in Ai then Lemma 10.1 would imply d(p, p ) <
d(p, q), contradicting the minimality of d(p, q). Therefore, Ai = {p} holds,
and Bi = {q} too, for the same reason. Thus, we can find a closest pair
p, q in the following way. For each of the m pairs (Ai , Bi ) in the WSPD,
we check, in constant time, if both sets Ai and Bi are singletons. If so,
we compute the distance between the two points, and determine a pair of
minimum value.
The interesting question is about the number m of pairs in a WSPD.
While the conditions in Definition 10.2 could be trivially met by
forming
all possible pairs of singleton sets, this would result in m = n2 = Θ(n2 )
many pairs, making the statement of Lemma 10.2 pointless. The following
result was shown in [177].
Theorem 10.2. There exists a well-separated pair decomposition of S that
consists of only O(n) many pairs. It can be constructed in time O(n log n).
Proof. (Sketch) Computing a WSPD is quite easy; the difficulty lies with
the analysis. First, a split tree T (S) is constructed that does not depend on
the separation constant s. One splits the bounding box R(S) of the given
point set S into two parts of equal size, by a hyperplane cutting orthogonally
through the midpoint of the longest edge of R(S). This hyperplane splits
S into subsets A and B, on which one recurs until only singleton sets are
left. Each node u is associated with the subset Su of all points stored in the
leaves of T (S) that have u as a predecessor.
Even though the resulting tree T (S) need not be balanced, it can
be constructed in time O(n log n), using the following techniques. In a
preprocessing step, the points in S are sorted by all coordinates, and
the resulting lists are pairwise doubly linked. When a box is split along
coordinate j, the corresponding list is searched for the split value, starting
from both ends simultaneously; this takes time proportional to the size of
the smaller subset. The building of T (S) is organized into O(log n) many
phases. In each phase, all subsets stored at the leaves of the current tree
are further split until their sizes are halved. One phase needs no more than
linear time.
Once T (S) is available, for each node with children v and w a procedure
findpairs(v, w) is invoked. If the subsets Sv and Sw are well-separated with
respect to s, the pair (Sv , Sw ) gets reported, and control returns. Otherwise
assume, without loss of generality, that R(Sv ) has a larger maximal
extension than R(Sw ) does. If vl , vr denote the children of node v in T (S),
two procedure calls findpairs(vl , w) and findpairs(vr , w) are carried out.
10.3. Well-separated pair decomposition
257
The crucial step of the analysis is in proving that only m = O(n)
many well-separated pairs are reported during all the recursive calls of
procedure findpairs. Since T (S) has only a linear number of nodes, only
O(n) many subsets of S can occur in the output. If, for a subset A, pairs
(A, B1 ), . . . , (A, Br ) are reported, the partner sets B1 , . . . , Br and their
bounding boxes must be pairwise separated by hyperplanes. A packing
argument shows that r is bounded by some constant. Hence, m = O(n)
holds. Since each recursion tree of an invocation of findpairs reports a wellseparated pair at each of its leaves, the total number of procedure calls is
linear, too.
While the split tree T (S) is clearly of linear size, the same holds for
the WSPD only if all sets Ai , Bi occurring in the decomposition are stored
implicitly, for example by a link to their corresponding nodes in T (S).
Although the number of pairs in the WSPDs just constructed is linear
in the size of point set S, the sum of the sizes of all sets Ai , Bi can be
quadratic. To overcome this disadvantage, a variant of the WSPD called
a semi-separated pair decomposition has been studied. The total size of all
subsets is now only O(n log n); see Abam et al. [3].
As an immediate consequence of Theorem 10.2 and Lemma 10.2, one
can use a WSPD to determine, in time O(n log n), the closest pair in a set of
n points. With extra effort, even more information can be extracted from
a WSPD, as the next theorem shows; cf. Subsection 8.1.2 for analogous
results in dimension 2.
Theorem 10.3. The k-nearest neighbors of each point in S can be
determined in time O(n log n + nk) by means of a well-separated pair
decomposition of S.
A main application of the WSPD is the construction of good spanners;
see Subsection 8.2.4 for definitions. The following surprising result has an
easy proof by induction on the ranks of the interpoint distances.
Theorem 10.4. Let (A1 , B1 ), . . . , (Am , Bm ) be a well-separated pair
decomposition of point set S with respect to a constant s > 4. For each
index i, where 1 ≤ i ≤ m, let ei be a line segment connecting a point of Ai
to a point of Bi . Then the collection of all edges ei forms an s+4
s−4 -spanner
of S.
That is, not only is the collection of edges ei a connected graph on S; we can
even find, for any two points p, q ∈ S, a path in this graph whose Euclidean
length does not exceed t · d(p, q), where t = s+4
s−4 . Clearly, t tends to 1 as s
grows to infinity. But increasing the precision comes at a cost, because the
258
Chapter 10. Alternative Solutions in Rd
constants hidden in the asymptotic estimates stated in Theorem 10.2 are
proportional to sd .
An O(n log n) algorithm for t-spanners can be used to efficiently
solve other problems like, e.g., approximating the minimum spanning tree
(Subsection 8.2.1) or approximating the dilation of graphs; see Narasimhan
and Smid [561] for these and many other results. They also discuss how to
apply the WSPD concept to metrics different from the Euclidean.
10.4. Post office revisited
Even though the well-separated pair decomposition presented in
Section 10.3 solves the closest pair and k-nearest neighbor problem in higher
dimensions quite efficiently, it does not offer a quick solution to the post
office problem introduced in Subsection 8.1.1.
Given a set S of n points in a space M equipped with a distance
measure, one wants to construct a data structure that supports the following
query: For an arbitrary query point q in M , report a point p ∈ S closest to q.
This problem is also called proximity, similarity, or nearest neighbor search
problem in the literature.
10.4.1. Exact solutions
In dimension 2 the Voronoi diagram’s locus approach provides an elegant
and optimal solution, cf. Subsection 8.1.1. In higher dimensions, Voronoi
diagrams can be too complex to be useful, as we discussed at the beginning
of this chapter.
One would hope for an alternative data structure in d-dimensional
Euclidean space Rd , that is of O(n) size and allows nearest neighbor
queries to be answered in time O(log n). Despite many efforts, no structure
has been found whose performance gets even close to these expectations.
Clarkson [225] and Meiser [540] proposed solutions that achieve logarithmic
query time but need space exponential in d. Alt and Heinrich-Litan [54]
considered the L∞ -norm and point sets randomly drawn from the unit
cube. Their algorithm needs only linear space but has an expected running
dn
time of roughly O( log
n + n), which improves on the brute force O(dn) time
bound if d is considered a variable.
It is widely conjectured that the ‘curse of dimensionality’ does not even
permit achieving a query time in O(n1−ε ) without using space exponential
in d. However, no lower bound that strong has been shown so far.
Some lower bounds have been established for the post office problem
in Hamming space {0, 1}d, where the distance between a and b equals the
10.4. Post office revisited
259
number of coordinates ai = bi . One has to assume that log n ≤ d holds
because the whole space contains only 2d many elements (corresponding
to the vertices of the unit hypercube in Rd ). A recent result by Pǎtraşcu
and Thorup states the following. If d = logc n, where c > 1, then any data
structure of size O(n polylog n) must have a super-logarithmic query time of
logc n
Ω
log log n
even for randomized algorithms; see [587], also for a definition of the cellprobe model used, and for a discussion of previous results by other authors.
One should observe that results for the Hamming space carry over to the
L1 -norm in Rd .
10.4.2. Approximate solutions
Since efficient exact solutions to the post office problem seem out of reach,
approximative algorithms have received a lot of attention. Let us assume
that, together with the point set S, an error bound ε > 0 is given. One
is interested in a data structure that provides the following approximate
answers. Given a query point q, let s denote a nearest neighbor of q in S.
Then a point s ∈ S must be reported such that
d(q, s ) ≤ (1 + ε) · d(q, s)
holds. Such a point s is called a (1 + ε)-nearest neighbor of q in S.
With this relaxation, the post office problem becomes tractable. Arya
et al. [74] obtained the following result.
Theorem 10.5. In time O(n log n) one can build a data structure of size
O(n) that allows approximate nearest neighbor queries in Rd to be answered
within time O(log n).
In [74] the constant in the query time bound is in O(d(1 + 6 dε )d ), while
preprocessing is independent of ε. The bounds on size and preprocessing
depend linearly on d. The construction works for all Lp -norms where p ≥ 1.
The data structure is based on a balanced box decomposition tree. Each
vertex corresponds to a rectangular annulus, i.e., to the complement of a
box within some larger, enclosing box.
Other approaches employed dimension reduction techniques as
described in Section 10.2. Kleinberg [468] obtained logarithmic query time
using space exponential in d. Indyk and Motwani [426] presented an
algorithm whose query time is logarithmic in n and only linear in d. The
space consumption, however, grows polynomially in n. They also introduced
260
Chapter 10. Alternative Solutions in Rd
Figure 10.3. Point location in equal balls (PLEB). Query point q is contained in the
balls of radius 2r centered at ci and cj , but in no ball of radius r.
an interesting reduction to a decision problem for point-location in equal
balls (PLEB). Given n balls of radius r centered at points c1 , . . . , cn , and
a query point q, is q contained in one of the balls? If so, return ‘yes’ and
a center cj whose r-ball contains q. Intuitively, the (approximate) distance
to the nearest neighbor of q can be determined by a binary search on a
sequence of exponentially growing radii; see Figure 10.3.
An approach of different flavor can be based on dynamic spanners;
see, e.g., Gao et al. [351] or Gottlieb and Roddity [375]. Suppose that a
(1 + ε)-spanner, G, of the point set S is available, as mentioned at the end
of Section 10.3. To obtain an approximate nearest neighbor in S for a query
point q, one can simply insert q into G, check all points of S adjacent to q
in G ∪ {q}, and report an adjacent point s of minimum distance to q. Let
π = (e1 , . . . , em ) denote the shortest path in spanner G ∪ {q} from q to a
true nearest neighbor s of q in S, with edges ei = (pi , pi+1 ). Then, indeed,
d(q, s ) ≤
m−1
d(pi , pi+1 ) ≤ (1 + ε) · d(q, s)
i=1
holds, because the first edge e1 must be of length at least d(q, s ). The
result in [375] provides dynamic spanners of constant degree; it implies
Theorem 10.5 for Euclidean space Rd and, more generally, for metric spaces
of constant doubling dimension. (Let λ denote the number of radius-r metric
balls needed to cover a metric ball of radius 2r. Then the doubling dimension
of the space is defined to be log2 λ.)
10.5. Abstract simplicial complexes
261
Har-Peled [395] introduced approximate Voronoi diagrams. Using
PLEB queries as described above, he computes, within time O(n log3 n),
a set of O(n log2 n) many regions. Each region C is either a hypercube or a
rectangular annulus, such that there is a point pC in S which is a (1 + ε)nearest neighbor to all points in C. These regions define a partition of Rd .
The hypercubes involved are stored in a compressed quadtree. Point-location
can be implemented to run in time O(log n).
This approach was further refined by Arya et al. [73]. They assign up
to t ≥ 1 many points of S to each region C, requiring that, for each z
in C, at least one of these points is a (1 + ε)-nearest neighbor of z. Now
the space consumption is only linear in n, thus providing another proof
of Theorem 10.5. While dimension d is considered fixed, some trade-off is
possible on the dependency of the error bound ε. For example, one can
achieve a query time and storage requirement of
n n
and O d−1 ,
O log
ε
ε
respectively, or use only O(n) space with a query time of
1
O log n + (d−1)/2 .
ε
Recently, Arya et al. [72] presented a unifying concept for the solution of
several proximity problems that matches these bounds without introducing
approximate Voronoi diagrams.
For further reading we refer to the surveys by Indyk [424] and
Clarkson [227], and to the recent monograph [396] by Har-Peled.
10.5. Abstract simplicial complexes
In many practical situations, the question of (re)constructing unknown
structures from data sample points arises. Of particular importance is
manifold reconstruction, where the objective is to preserve the topological
type, and to approximate the geometry. Applications stem from diverse
areas, like visualization, pattern recognition, signal processing, neural
computation, and others; see, e.g., Boissonnat et al. [144, 149] for references.
Subcomplexes of the Delaunay triangulation (cf. Subsections 6.6.2 and 8.2.2)
offer elegant solutions in low dimensions, especially the Delaunay simplicial
complex restricted to a surface, as is described in detail in Boissonnat and
Teillaud’s book [149].
Sample points in parameter space can be of high dimension, though. As
the size of the Delaunay simplicial complex DT(S) grows too fast, methods
whose algorithmic complexity depend only on the intrinsic dimension of the
262
Chapter 10. Alternative Solutions in Rd
manifold (rather than on the dimension of the ambient space) are sought.
To this end, more ‘light-weight’ structures have been investigated, which
we will describe to some extent in the present section.
Given a finite set S of elements, an abstract simplicial complex for S
is a collection A of subsets of S that contains, for each member X of A,
all possible subsets of X as well. The members of A are called abstract
simplices. For example, if S is a set of n points in d-space, the subsets of
points which span simplices in DT(S) form an abstract simplicial complex.
However, not every abstract complex is geometrically realizable.
A prominent abstract simplicial complex is the Vietoris–Rips complex,
often just called Rips complex, Rα (S), of a set S of points in d-space. Its ksimplices correspond to the (k + 1)-tuples in S whose points are at pairwise
distance less than α. See Figure 10.4 for an example. This structure was
introduced by Vietoris [690] and Rips (see Gromov [382]) as a tool for
studying hyperbolic groups. Ghrist and Muhammad [362], and others, have
proposed Rips complexes as a light-weight representation of the topological
structure of high-dimensional data. Chambers et al. [187] address basic
homotopy problems in Rips complexes in the Euclidean plane. For example,
they describe efficient algorithms for determining whether a given cycle is
contractible in the Rips complex of a finite planar point set.
The nerve of a finite set B of balls in d-space is the collection of
all subsets of B with non-empty common intersection. It is an abstract
simplicial complex; see Edelsbrunner [302] for its various properties. In
particular, the Čech complex, Cα (S), of a point set S is the nerve of the set
of (congruent) balls of radius α around the points in S. (Note the similarity
of these complexes to the (weighted) α-shapes defined in Subsection 6.6.2.)
Čech complexes are closely related to Rips complexes, which is made explicit
in the following lemma.
Figure 10.4. This Rips complex consists of ten 0-simplices (the defining points),
seventeen 1-simplices (the shown edges), twelve 2-simplices (spanned triangles), five
3-simplices (spanned ‘tetrahedra’), and one 4-simplex (the complete graph K5 ). The
distance α is visualized by a disk of this diameter.
10.5. Abstract simplicial complexes
263
Lemma 10.3. For any finite point set S, we have C α2 (S) ⊆ Rα (S) ⊆
Cα (S).
Another related class of abstract simplicial complexes has been
considered in the literature. These complexes are defined by two sets of
points, namely, the underlying vertex set S, and some set P of so-called
witness points. We say that a k-simplex spanned by S is witnessed by a
point p if its vertices are among the k + 1 nearest neighbors of p in S. The
witness complex for S and P now contains all simplices spanned by S that
are witnessed by some point in P . More generally, for fixed α ≥ 0, the
complex Wα (S, P ) contains all simplices spanned by S, whose vertices v lie
within distance d(v, p) + α from some witness p ∈ P . We have the following
inclusion relation:
Lemma 10.4. If each point in S lies within distance β from some point
in P, then C α−β (S) ⊆ Wα (S, P ) holds, for all α > β.
2
Already for the standard case α = 0, the witness complex W0 (S, P )
is a versatile tool, because the choice of the witness set P can be used to
control the selection of the simplices connecting the vertex set S. Based
on the empty-sphere property of the Delaunay simplicial complex, de Silva
and Carlsson [259] prove that a k-simplex ∆ spanned by S is included in
DT(S) if ∆ and all its faces have a witness in P . Consequently, W0 (S, P )
is an approximation of DT(S) and, in the limit when P covers the entire
d-space, both complexes coincide.
For reconstructing manifolds, restricted versions of DT(S) have to be
considered. Given a submanifold M in d-space, the Delaunay complex of S
restricted to M , for short DTM (S), consists of all the faces of DT(S) whose
dual Voronoi faces intersect M .
Several authors consider the subcomplex DTM (S) in relation to the
witness complex W0 (S, P ), for the particular case where the points in S are
taken from P . Guibas and Oudot [392] report that, if badly shaped simplices
(so-called slivers) occur, then W0 (S, P ) is not included in DTM (S), and
homotopy equivalence is lost, even for dense sample sets S. Informally
speaking, a sliver is a flat simplex all of whose edges are still long. In [392], a
strategy is proposed for selecting S in order to keep homotopy equivalence,
in the case where M is of exactly one dimension lower than the ambient
space. Attali et al. [84] generalize this result to arbitrary dimensions of the
submanifold M , and show that if M is a smoothly embedded surface, then
a sufficiently fine sample S of M will make W0 (S, P ) approximate DTM (S).
Boissonnat et al. [144] assign weights to the points in S, in order to get
rid of slivers, building on results in Cheng et al. [201]. If w(p) ≥ 0 denotes
264
Chapter 10. Alternative Solutions in Rd
the weight of a point p ∈ S, then the condition
max
p=q ∈ S
1
w(p)
≤
d(p, q)
2
ensures that no cell in the resulting power diagram of S (see Section 6.2) is
empty, and that each point in S stays in its power cell.
(This is one more possibility of defining so-called self-centered
diagrams, different from and opposed to the concepts of a centroidal Voronoi
diagram or power diagram in Subsections 9.3.4 and 6.4.3, and of a selfcentered simplicial complex in Section 6.1. Self-centered power diagrams
whose cells contain their owning sites find further applications, for example
in data visualization; see Andrews et al. [62] and Granitzer et al. [377].)
Self-centering weights can be found such that, in the dual regular
simplicial complex for S (see Section 6.3) no slivers do occur, and the
witness complex is included in this complex, and is homeomorphic to the
underlying manifold, for suitable sample sets S. The algorithm in [144] runs
in O(c1 (δ) · n2 ) time, where n = |S| and c1 (δ) is a constant depending only
on the intrinsic dimension δ of the manifold, though super-exponentially.
2
More precisely, c1 (δ) = 2O(δ ) .
A thorough study of the properties and use of the various abstract
simplicial complexes defined above is given in Chazals and Oudot [193].
They obtain new results on Rips, Čech, and witness complex filtrations
(i.e, nested sequence of complexes of the same type). Based on such data
structures, an algorithm is developed that, for smooth and sufficiently
densely sampled manifolds, derives the homology in time O(c2 (δ) · n5 ),
δ
where c2 (δ) = O(835 ). Topological estimation can be performed at a userdefined scale. Details are involved and go beyond the scope of this section.
Let us just state the following nice result, concerning regular simplicial
complexes and witness complexes.
Given some subset M of d-space, a finite point set S ⊂ M is said to be
an ε-sample of M if every point of M lies within distance ε of S. Moreover,
we call S an ε-sparse sample if its points guarantee a minimum distance
of ε from each other.
Lemma 10.5. Let M be a compact manifold in d-space. Assume that
P ⊂ M is an ε1 -sample of M, and that S ⊂ P is an ε2 -sample of P which is
ε2 -sparse. For some weighting of S, consider the regular simplicial complex
of S restricted to M, for short RCM (S). Then, if the weighting fulfills the
maximum condition stated above, we have RCM (S) ⊆ Wα (S, P ) whenever
α ≥ 83 (ε1 + 14 ε2 ).
10.5. Abstract simplicial complexes
265
If S is an unweighted point set (with w(p) = 0 for all p ∈ S) then
the maximum condition on weights is automatically fulfilled. Moreover,
RCM (S) becomes the restricted Delaunay simplicial complex DTM (S), and
Lemma 10.5 can be strengthened to DTM (S) ⊆ Wα (S, P ), for α ≥ 2 ε1 .
We mention that the concept of witness complexes is different from
the concept of witness Delaunay graphs [70] discussed in Subsection 8.3.2,
which is based on the empty-circle property.
Very recently, Sheehy [644] provided a (computationally more
tractable) approximation method for Rips complex filtrations. Recall that
the Rips complex, Rα (S), of a set S of n points in d-space consists of
all simplices spanned by S whose diameter is less than α. The so-called
persistence diagram of the Rips filtration represents the changes in topology
that correspond to the changes in Rα (S), when the parameter α is varied.
A simplicial complex of size only O(n) now can be constructed in time
O(n log n), such that its persistence diagram is a good approximation to
that of the Rips filtration. The constants depend only on the doubling
dimension (which is O(d) for Euclidean d-space) and, of course, on
the accuracy of the approximation. The main tools used are metric
spanner graphs and the efficient data structures for computing them; see
Section 10.3.
The paper [644] also gives a nice overview on the topic of abstract
simplicial complexes, with various further references.
Finally, we mention that a new method for storing and simplifying
simplicial complexes has been proposed recently, in Attali et al. [85]. The
encoding seems to be particularly effective for Rips complexes in higher
dimensions.
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Chapter 11
CONCLUSIONS
In this book, we have strived to present a reasonably complete and timely
treatment of Voronoi diagrams and their relatives, from the point of view of
a computational geometer. We hope that this book will be of use and help to
people in various fields of science and engineering where space partitioning
structures and algorithms for computing them are of importance. We also
wish that we had conveyed the elegance and beauty of Voronoi diagrams —
be it as a structure per se, or concerning the methods for analyzing and
constructing them — and that interested readers from education, design,
and art will also benefit from our expositions.
At the time of this writing, an internet search for ‘Voronoi’ generates a
list of about 7 · 105 articles, more than a single survey could cover. We have
tried to focus on structural and algorithmic aspects of Voronoi diagrams,
more than on their applications in other sciences. But even in this area,
not every important and interesting topic could be given the attention it
deserves.
11.1. Sparsely covered topics
Let us mention a few noteworthy aspects of Voronoi diagrams that have
not been treated in detail in the present book.
A significant stream of research concerns the stochastic properties of
Voronoi diagrams for randomly distributed sites. Motivation stems from
their relevance in the natural and social sciences. The interested reader is
referred to Chapters 5 and 8 of Okabe et al. [571] for a comprehensive
treatment. Also, the survey paper [93] could be consulted for a brief
overview. In the present book, randomization mainly affects the choices
made by an algorithm during execution, whereas the input is usually
considered fixed, but arbitrary.
267
268
Chapter 11. Conclusions
Only in a minority of cases, expected quantities that result from
assuming some underlying probability distribution of the input sites have
been treated. Examples are the expected runtime of a Voronoi diagram
construction algorithm for n sites uniformly distributed in the square, or
the expected combinatorial size of a high-dimensional Delaunay complex
whose sites are restricted to lie on a lower-dimensional manifold. We
decided not to include other stochastic aspects, like metric properties of
Voronoi regions (e.g., expected perimeter, or variance of their volume), or
combinatorial quantities (e.g., expectations for the exact number of Voronoi
edges and facets), as their relevance to computational geometry is less
direct.
In view of the broad scope of applications of Voronoi diagrams, where
the input data to be processed tend to rapidly increase in quantity, fast
construction is an important issue. Parallel algorithms are a tool to reach
this goal. We have mentioned a few, but did not detail and analyze parallel
methods in this book. In many cases, divide & conquer methods for
computing Voronoi diagrams and Delaunay triangulations lend themselves
to efficient parallelization, but sometimes also the plane sweep technique
does. We refer the reader to the handbook chapter [372] by Goodrich for an
exposition of this material. Also, the chapter by Atallah and Chen devoted
to parallel computational geometry, in the handbook [82], might partially
cover this gap.
There are two radically different ways of representing a Voronoi diagram
in a computer: By exact vector geometry, which we have been dealing with
in this book, opposed to a representation as a binary image or pixel map,
i.e., an integer square grid in Z2 each of whose cells represents a pixel, or
a voxel for Z3 . We have mentioned such digital (pixel) Voronoi diagrams,
or Voronoi diagrams on the grid, only sporadically at important occasions.
Such diagrams can be computed either directly on the pixel map, or by any
exact geometric algorithm (see Chapter 3) followed by pixel extraction.
They play an important role in image processing, computer vision and
graphics, pattern matching, and also in various other fields where the
discrete medial axis (often called the medial axis transform, MAT) of a
planar shape has been applied with advantage over the years.
Pixel Voronoi diagrams can be trivially constructed in O(m2 n) time
on an m × m pixel map where n individual pixels are marked as sites,
by assigning to each pixel the ‘color’ of its owning site in O(n) time.
In principle, this works for any distance function and, except its reliable
calculation, no further geometric or structural properties are needed.
Enhanced information is comprised in the (Euclidean) distance
transform, EDT, where each pixel holds the (squared Euclidean) distance
11.2. Implementation issues
269
to its closest site. The EDT can be computed in optimal time O(m2 )
with several methods. We refrain from details, which can be found in the
comprehensive survey article by Fabbri et al. [333], and also in the overview
of skeleton extraction methods in Siddiqi and Pizer [654].
The Voronoi diagram on the grid should not be confused with the
Voronoi diagram for a lattice, in Subsection 7.1.2. There all the lattice
points are sites, rather than just a few, and the diagram is a cell complex
of congruent polytopes (prototiles) in Rd .
Finally, let us admit that — while a host of applications of the Voronoi
diagram have been presented — applications in the natural and social
sciences have not been described in great detail in most cases. This would
have had to include a thorough treatment of stochastic properties of Voronoi
diagrams as well. We are aware that the appearance of Voronoi diagrams in
natural phenomena is an indubitable indication of their inherent elegance
and their usefulness, also in other areas of science. The book by Okabe
et al. [571] puts strong emphasis on these important applications, which
led us, being no experts in these areas, to encourage the reader to consult
this book instead.
11.2. Implementation issues
Readers who want to apply Voronoi diagrams in practice might be
interested in implementation aspects. Since implementing a geometric
algorithm is an important and demanding task, we include some comments
on this topic. For example, numerical problems are a big (but not the only)
critical issue in this context, in computational geometry and in geometric
and scientific computing in general. We have already given a few citations
of relevant work at appropriate places in the book.
Are two straight lines exactly parallel, or do they intersect at a
far remote point? Can we be sure that this line segment touches the
circumference of that circle? Decisions based on queries of this kind might
critically influence the subsequent control flow of a geometric algorithm —
and with it — its accuracy or even worse, its correctness.
Theoretical work on geometric algorithms is often based on the
idealistic, and sometimes unrealistic, assumptions that (i) inputs are in
general position, and (ii) that real number arithmetic is exact and can be
performed in constant time.
Assumption (i) is used to rule out ‘degenerate’ input configurations
that deviate from the ‘generic’ case and tend to complicate matters. For
example, if we assume that no three among n points in set S lie on a line, and
no four points on a circle, then the Delaunay tessellation DT(S) is always a
270
Chapter 11. Conclusions
triangulation of S, nice to look at and easy to construct incrementally;
see Section 3.2. That the configurations ruled out are contained in a
lower-dimensional subset of the configuration space R2n of the point
coordinates seems to be justification for ignoring them.
However, in a VLSI application, input sets may consist of grid points
that fail to be in general position. Consequently, we need to implement
algorithms in such a way that they can cope with all inputs possible. There
are different ways to proceed.
Starting with an algorithm A that works for input in general position,
one could extend A into an algorithm that can handle degenerate input,
too. Sometimes this can be done in an organic way, without implementing
complicated case analysis. For example, the divide & conquer algorithm
presented in Section 3.3 can be extended to gracefully handle Voronoi
vertices of degree larger than 3; see [461]. It computes the Voronoi
diagram V (S) for arbitrary n-point sets S in the plane, hence DT(S) by
dualization.
Another elegant approach is based on the idea that moving the points
in S by an infinitesimal amount will remove degeneracies like the ones
mentioned above. Thus, one first applies a perturbation to the input in
such a way that general position is achieved. Then algorithm A is run
on the perturbed input. One difficulty is in recovering the true output on
input S, from the output which A has computed on the perturbed input.
See Seidel [632] for a unifying presentation of the perturbation approach, a
discussion of various methods, and for further references.
Now let us turn to assumption (ii) on perfect real arithmetic. Most
programming languages offer standardized fixed or floating point reals,
with a predefined number of digits, fixed size integers, and integers
of (potentially) unbounded length. While floating point arithmetic is
hardware-supported and therefore quite fast, it is prone to rounding errors.
Computing with long integers, on the other hand, is exact but slow.
A floating point filter approach tries to avoid long integer computations,
using fast floating point arithmetic in combination with error bounds; see
Mehlhorn and Näher [536].
If the input objects are specified by floating point or rational
coordinates, they can be exactly represented by the above number types.
As soon as an algorithm generates new geometric objects from given
ones, their coordinates need no longer be rational numbers, however. For
example, the plane sweep algorithm presented in Section 3.4 generates
Voronoi vertices as intersection points of parabolas. Their coordinates
involve square roots. Bisectors of more general curved objects, as discussed
in Section 5.5, can be of larger algebraic degree; see Emiris et al. [319] for
11.2. Implementation issues
271
the case of ellipses. In general, one needs to find the real zeros of polynomial
equations
f (x) =
n
ai xi
i=0
with rational coefficients ai . If the degree n exceeds 4, the zeroes of f can
in general not be expressed by nested radicals in the numbers ai . Instead,
the real zeroes of f must be isolated by rational intervals, and then be
numerically approximated. The problem becomes even more challenging
if the coefficients ai themselves are algebraic numbers, only available by
approximation.
We recommend the book by Boissonnat and Teillaud [149] that offers
several solutions on how to deal with such problems, especially in the
context of curves and surfaces arising in the construction of Voronoi
diagrams and arrangements, and the book by Yap [713] for the algebraic
background.
Both issues, degeneracy and accuracy, are interweaved. Namely, many
geometric algorithms proceed by evaluating geometric predicates. For
example, let InCircle(p, q, r, s) return a value < 0, = 0, or > 0 if point s
lies inside, on, or outside of the circle defined by p, q, r, respectively.
Value 0 characterizes a degenerate situation, and the other two cases require
sufficient accuracy to return the proper result. Controlled perturbation is a
new approach that uses perturbation to avoid degeneracies and to bypass
precision problems at the same time; see Mehlhorn et al. [537].
Over the years, a lot of geometric software has been developed, including
implementations of Voronoi diagrams and Delaunay triangulations.
Large algorithm libraries are LEDA (Library of Efficient Data types
and Algorithms; Mehlhorn and Näher [536]) and CGAL (Computational
Geometry Algorithms Library [185]). The interested reader may also consult
the handbook chapter [452] by Kettner and Näher, and the books by
Boissonnat and Teillaud [149] for curves and surfaces in CGAL and, very
recently, Fogel et al. [341] for surface arrangements in CGAL. (Recall
from Subsection 7.5.1 that computing envelopes and arrangements of
Voronoi surfaces is a powerful and flexible approach to constructing Voronoi
diagrams.) An introduction to geometry software outside these libraries has
been given in a handbook chapter by Joswig [434].
On the ‘piecewise linear side’, the book on computational geometry
in C by O’Rourke [572] offers various ready-to-use pieces of code in the
programming language C, covering convex hulls in 2D and 3D, Delaunay
triangulations, and planar polygon treatment.
272
Chapter 11. Conclusions
To visualize a Voronoi diagram seems a somewhat easier task than to
fully construct it. There are a multitude of interactive applets available on
the internet, that allow to experiment with many variants of the fascinating
structure we have presented in this book.
11.3. Some open questions
The body of literature on Voronoi diagrams is vast, and many of the
arising theoretical and practical problems have found elegant and efficient
solutions. Naturally, new questions have emerged from research conducted,
and quite a few problems eluded satisfactory settlement until today.
Let us list — out of the various interesting open problems on Voronoi
diagrams we have presented in this book — a few that seem most important
in view of their basic nature and practical relevance. They concern both
the combinatorial complexity of (seemingly) simple structures and their
efficient algorithmic construction.
Among the few algorithmic problems for planar Voronoi diagrams
which lack a worst-case optimal solution is the construction of the order-k
Voronoi diagram for a fixed value of k. Only close-to-optimal algorithms
do exist (Subsection 6.5.3). And notably, the usually very successful plane
sweep approach, which has been applied to this problem recently, also
leads to a runtime of only O(k 2 n log n); see [585]. An easy lower bound
is Ω(kn + n log n).
Placing an additional site such that its region area is optimally large
when inserted in an existing Voronoi diagram is still an unsolved question
(Subsection 9.2.1). Under which restrictions, other than Theorem 9.1, does
the question become easier?
Analyzing the Voronoi diagram of moving point sites (Section 9.1)
seems to be among the hardest open problems in this area. Recent
achievements [611], though, kindle the hope for a near-quadratic upper
bound on the number of flips in the kinetic Delaunay triangulation for
certain site trajectories. In particular, the case of sites moving at constant
speed on straight lines may have such a solution within reach.
Despite its importance in practice, the minimum spanning tree in
3-space (Subsection 8.2.1) has eluded efficient construction so far. Also, the
existing subquadratic solutions [11, 710] are far from easy implementation.
A partial answer is the available efficient expected-time and approximation
algorithms (cf. also Chapter 10). Is there a practical and worst-case efficient
algorithm for computing exact minimum spanning trees in 3-space?
A theoretical challenge is the analysis of the maximal size of the medial
axis of a nonconvex polyhedron in 3D (Section 6.6). The known best upper
11.3. Some open questions
273
and lower bounds are more than an order of magnitude apart. The same
is true for the Voronoi diagram of straight lines in 3-space. Even worse is
the situation for the straight skeleton (Section 5.3). This seemingly simpler
for piecewise linear skeleton is notoriously difficult to construct, even in the
plane. Also, its size in the three-dimensional case is unclear, apart from a
near-quadratic lower bound [115].
Can any two tetrahedrizations on the same point set in 3D be
transformed into each other by bistellar flips? Apart from theoretical
interest, flip graph connectedness for tetrahedrizations would ‘legalize’
certain algorithms currently in use, which try to turn a tetrahedral mesh
into the Delaunay tetrahedrization (Subsection 6.3.3).
Finally, an efficient exact solution for the post office problem in d
dimensions is still outstanding (Section 10.4). The hope for a close-toO(log n) solution is sparse, however, in view of the discouraging lower bound
for Hamming spaces.
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INDEX
abstract inverse Voronoi diagram, 174
abstract order-k Voronoi diagram,
174
abstract simplicial complex, 202, 216,
262
abstract Voronoi diagram, 57, 168,
181
Ackermann’s function, 61, 112, 143
acyclicity property of complexes, 45,
80
additively weighted diagram, 86, 113,
121, 145, 154, 171
additively weighted distance, 154
admissible bisector system, 169, 181
affine Voronoi diagram, 86, 165
air-lift metric, 144, 148, 176
all nearest neighbors problem, 186,
248
alpha-shape, 45, 88, 119, 200
angular bisector, 53, 55
angular Voronoi diagram, 159
angularity, 42, 77, 105, 138
anisotropic Delaunay triangulation,
163
anisotropic Voronoi diagram, 159, 163
annulus, 194, 259
anti-center of a graph, 248
Apollonius circle, 157
Apollonius model, 86, 158
approximate nearest neighbor, 259
approximate Voronoi diagram, 261
approximation algorithm, 39, 99, 115,
198, 199, 216, 240, 245, 259
area of union, 119
arrangement of (hyper)planes, 106,
114
arrangement of hypersurfaces, 168
arrangement of pseudo-lines, 250
aspect ratio, 234
associahedron, 90
augmenting path, 98
balanced fractional clustering, 102
basin of attraction, 242
beta-skeleton, 202, 205
bi-criteria spanning tree, 198
biarcs, 70
bichromatic 2-center problem, 193
bichromatic closest pair, 188, 197
binary medial axis, 116, 268
bipartite graph, 210
bisecting curve, 168
bisector, 7, 75, 86, 115, 131, 168
bisector system, 169
bistellar flip, 78, 90
boat-sail distance, 156
bottom-up divide & conquer, 27, 67
bounded set, 8
bounded Voronoi diagram, 62, 65
bounded-degree Delaunay, 189
bounded-degree spanner graph, 205,
260
329
330
bounding box of point set, 254
box decomposition tree, 259
Bregman divergence, 86
bucketing technique, 23, 79
calibration figure (Eichfigur), 130
canonical Voronoi insertion, 191
cardinality of a set, 18
Cauchy sequence, 146
Cech complex, 262
cell complex, 75, 87
cell-tuple data structure, 82
center of a graph, 248
centrally symmetric polytope, 129
centroid, 44, 110, 212, 244
centroidal power diagram, 103
centroidal Voronoi diagram, 244
certificate of Delaunay edge, 227
chordale of circles, 81
circle packing, 191
circle-based β-skeleton, 202
circular arc shape, 70
circularly separable clustering, 214
circumcircle, 12, 19, 41, 43, 139, 208
circumradius, 129, 139
city metric, 176
city Voronoi diagram, 55, 176
clearance of a robot, 217
closed set, 8
closest covered set diagram, 112
closest pair problem, 188, 248, 255,
257
closest-polygon Voronoi diagram, 172
closure of a set, 149
cluster selection, 214
cluster Voronoi diagram, 112, 215
clustering, 93, 103, 160, 211
coarseness of a triangulation, 43, 76
collision detection, 58, 217
collision-free path, 218
colored 2-center problem, 193
colored closest pair, 188
compact order-k Voronoi diagram,
185
compact Voronoi diagram, 55, 185,
220
compatible triangulations, 64
competitive facility location, 233
Index
complete graph, 76, 195, 205, 210, 248
complete-linkage clustering, 215
composite metric, 148
configuration space, 89, 98, 220, 270
conflicting triangle, 19, 21, 80
conforming Delaunay triangulation,
64
constrained Delaunay
tetrahedrization, 93
constrained Delaunay triangulation,
63, 93, 192, 196, 205
constrained minimum spanning tree,
196
constrained order-k Delaunay
triangulation, 208
constrained regular triangulation, 93
constrained simplicial complex, 93
construction history, 22, 27, 54, 79,
111, 184
continuous Dijkstra technique, 65,
127, 180
controlled perturbation, 271
convex distance function, 130, 141,
147, 171, 174, 219, 241
convex graph, 247
convex hull, 10, 16, 25, 32, 51, 84,
104, 109, 200, 251
convex hull algorithms, 85
convex hull of spheres, 120
convex hull-disjoint, 100
convex partition, 35, 206
convex polygon, 8, 10, 53, 60
convex polyhedron, 32, 83
convex polytope, 84, 89, 96
convex position of point set, 10, 13,
16, 32, 54, 90, 209, 231, 237
convex set, 8
covering, 238
cross-polytope, 143
cross-section of Voronoi diagram, 86
crossing-free geometric graph, 196
crust of sample points, 201
crystal growth Voronoi diagram, 162,
240
crystallographic group, 129
Davenport-Schinzel sequence, 30, 227
decomposition, 9
Index
decomposition lemma (medial axis),
72
degree of a vertex, 9, 20, 199
Delaunay flip, 19, 90
Delaunay for sites on surfaces, 128
Delaunay neighbor, 187
Delaunay refinement, 191
Delaunay repair, 228
Delaunay simplicial complex, 76, 128,
210, 261
Delaunay stabbing problem, 236
Delaunay structure for graphs, 248
Delaunay subcomplex, 261
Delaunay tessellation, 12
Delaunay tessellation in 3-space, 76
Delaunay tetrahedrization, 76
Delaunay tree, 22, 111
Delaunay triangulation, 12
Delaunay triangulation on surfaces,
128
depth-first search, 37
diameter of a graph, 73
diameter of a point set, 140, 192
digital Voronoi diagram, 15, 162, 268
Dijkstra’s algorithm, 127, 220, 247
dilation of a graph, 205, 209, 258
dilation of a path, 152
dimension reduction, 252, 259
directed acyclic graph, 22
Dirichlet tessellation, 3
discrepancy, 226
discrete group of motions, 125
discrete medial axis, 116, 268
discrete straight skeleton, 61
distance cone, 153, 155, 156, 161
distance graph, 164
distortion of an embedding, 253
divide & conquer, 22, 24, 50, 66, 71,
113, 133, 136, 152, 172, 270
domain of action, 2
doubling dimension, 260, 265
doubly connected edge list, 15, 31
dual structure, 3, 12, 16, 51, 136, 263
duality transform, 84, 109
dynamic data structure, 225
dynamic k-nearest neighbor
searching, 226
dynamic point-location, 226
331
dynamic
dynamic
dynamic
dynamic
post office problem, 185, 226
programming, 203
spanner graph, 260
Voronoi diagram, 22, 27, 111
ear of a polygon, 105
ear-clipping algorithm, 105
edge event (straight skeleton), 56
edge exclusion property, 206
edge flip, 19, 42, 137
edge graph of Voronoi diagram, 8, 67
edge inclusion property, 206
ellipsoid, 174, 193
empty neighborhood graph, 202, 204
empty-circle property, 12, 35, 63, 136,
138, 227
empty-sphere property, 76, 78, 263
epsilon-closeness problem, 17, 188
epsilon-net, 226
epsilon-sample, 264
equiangularity, 42, 77
equilibrium state, 39, 235, 238
equipartition of a convex body, 96
Euclidean distance, 7, 131
Euclidean distance transform, 268
Euler’s formula, 11, 49, 177
evenness property of point set, 251
exchanging flip, 90
expected storage requirement, 22
extreme point in a set, 10, 12, 104
face-to-face complex, 75, 82
facet of a cell complex, 75, 82
facet-edge data structure, 76
facility location problem, 35, 189,
229, 248
farthest-polygon Voronoi diagram, 52,
172
farthest-segment Voronoi diagram,
51, 106, 172
farthest-site abstract Voronoi
diagram, 174
farthest-site Delaunay complex, 105
farthest-site Delaunay triangulation,
105, 200
farthest-site graph Voronoi diagram,
247
332
farthest-site Voronoi diagram, 51, 85,
103, 113, 160, 192, 200
fat object, 119
fatness of a triangulation, 44
Fermat point, 197, 213
Fibonacci heap, 247
filtration of cell complexes, 264, 265
finite element methods, 5, 43, 93
finite set of orientations, 115, 133,
143, 181
fixed point, 240, 245
fixed tree theorem, 196
fixed-share decomposition, 60
flat torus, 128
flip distance, 42, 91
flip graph, 209
flipping, 19, 42, 78, 90–92, 128, 137
floating point filter, 270
forest, 186, 196
Fréchet distance, 66
fractal set, 243
fractional clustering, 102
framework statics, 39
free space of a robot, 220, 222
fundamental domain, 126, 129
Gabriel graph, 141, 202, 205, 211, 250
general position of point set, 12, 15,
48, 49, 76, 117, 119, 141, 186, 269
generalized Dirichlet tessellation, 81
geodesic distance, 64, 206
geodesic path, 123, 147
geodesic Voronoi diagram, 64, 123,
162, 175, 176
geometric predicate, 271
geometric software, 271
geometric transformation, 28, 31, 83,
109
gerrymander problem, 36
gradient method, 98, 99
graph, 8
graph drawing, 160, 209, 211
graph Voronoi diagram, 246
grass fire interpretation, 154
greedy algorithm, 196
greedy triangulation, 45, 203, 206
growth model, 35, 57, 144, 161, 177,
240
Index
Halley’s method, 243
Hamiltonian cycle, 198, 209
Hamming space, 258
Hausdorff distance, 66, 112, 215
Hausdorff Voronoi diagram, 112, 171
Helly’s theorem, 175
higher-order Voronoi diagram, 103
highway hull, 181
histogram, 63
homothet, 137, 143
Hotelling game, 235
hull-disjoint, 100, 112, 214, 215
hyperbolic Delaunay complex, 124
hyperbolic geometry, 124
hyperbolic Voronoi diagram, 86, 124
hypercube, 95, 143, 259
hyperplane arrangement, 106
hyperrectangle, 254
imprecise point sets, 23
in-front/behind relation, 45, 80
incircle test, 23, 27
incremental insertion, 18, 50, 77, 97,
113, 128, 136, 152, 172, 181, 184
incremental search, 23
independent edge set, 91, 208
independent vertex set, 237
inradius, 44, 129
inter-cluster measure, 212, 214
interdistance enumeration problem,
189
interference pattern, 55, 154
interior of a set, 8
intersection of balls, 82
intra-cluster measure, 212
intrinsic dimension of manifold, 261,
264
inversion transform, 32
inward graph Voronoi diagram, 247
isothetic, 80, 113, 176
iterated logarithm, 51
Johnson–Mehl model, 2, 86, 155
Jordan curve, 145, 150, 170, 171
Jordan matrix, 165
k closest pairs problem, 188
k-center problem, 193, 213
Index
k-centroid problem, 212, 246
k-clustering, 212
k-facet of a point set, 109
k-flat, 114
k-level greedy triangulation, 204
k-level in an arrangement, 108
k-locally Delaunay graph, 248
k-locally Gabriel graph, 250
k-means clustering, 212, 246
k-median, 189
k-nearest neighbor problem, 103, 185,
226, 257
k-neighborhood in graphs, 248
k-sector curve, 241
k-set of a point set, 108
k-simplex, 87
k-triangle of a point set, 109
Karlsruhe metric, 148
kinetic Delaunay triangulation, 227
kinetic spanner graph, 206
kinetic Voronoi diagram, 27, 50, 227
Klein model of hyperbolic geometry,
124
Kruskal’s algorithm, 196, 214
Kullback–Leibler divergence, 86
L1 -norm, 131, 176, 197, 206, 241, 259
L1 -Voronoi diagram, 177
L∞ -medial axis, 59, 61, 143
L∞ -norm, 59, 131, 197, 214, 258
Lp -higher order Voronoi diagram, 109
Lp -norm, 131, 142, 259
Lp -Voronoi diagram, 133
Laguerre-Voronoi diagram, 81
largest empty circle problem, 103,
190, 248
largest empty circum-shape, 139
lattice Delaunay polytope, 129
lattice prototile, 129
least-squares clustering, 94, 213
least-squares fitting, 101
least-squares matching, 101
level in an arrangement, 106
lifting map, 27, 31, 84, 92, 109, 110
light triangulation edge, 203
line skeleton, 59
line Voronoi diagram, 48
linear axis of polygon, 59
333
linear programming, 37, 192, 194
linearization of a Voronoi diagram,
54, 86
linearly separable clustering, 214
Lipschitz function, 253
Lloyd’s method, 245
LMT-skeleton, 204
locally Delaunay, 41, 51, 79, 138
locally equiangular, 42
locally minimal triangulation, 44, 204
locally regular facet, 90
locational optimization, 4
locus approach, 183, 185, 258
Lombardi drawing, 161
Lorentz transformation, 166
low-dimensional embedding, 252
lower convex hull, 32, 84
lower envelope, 33, 99, 112, 114, 153,
168, 173
lune-based β-skeleton, 202
m-star-shaped region, 150
m-straight path, 146, 171
Möbius transformation, 161
Manhattan distance, 131
manifold reconstruction, 261
matching, 101, 198, 237
maximal inscribed ball, 117
maximal inscribed disk, 68, 71
maximal outerplanar graph, 210
maximal territory diagram, 242
maximized Voronoi region, 229
maximum norm, 131
Maxwell’s theorem, 39
Maxwell–Cremona theorem, 40
medial axis, 2, 48, 50, 53, 58, 67, 71,
114, 143, 202, 223
medial axis transform, 71, 117, 268
medial ball representation, 118
merge chain, 24, 134, 152, 172
mesh generation, 43, 63, 93, 191
mesh improvement, 159, 191
metric, 144
min–max Voronoi diagram, 112
minidisk algorithm, 193
minimal Steiner tree, 197, 246
minimization diagram, 33, 168
minimum convex partition, 207
334
minimum convex Steiner partition,
207
minimum diameter spanning tree, 197
minimum length matching, 198
minimum spanning tree, 139, 187,
189, 195, 202, 205, 214, 258
minimum-cost flow problem, 98
minimum-cost load balancing, 155
minimum-weight triangulation, 44,
202, 206
minimum-width annulus, 194
Minkowski difference, 220
Minkowski norm, 131
Minkowski sum, 74, 140, 221
Minkowski theorem, 96
mitered offset, 57, 74
mixed weighted distance, 160, 213
molecular modeling, 117, 120
moment curve, 251
Monge–Kantorovich problem, 95, 155
monotone curve, 24, 147
monotone polygon, 57, 58
morphing of shapes, 120, 196
Moscow metric, 148
motion planning problem, 74, 216
motorcycle graph, 58
multiplicatively weighted diagram,
86, 156, 172
multiplicatively weighted distance,
156
n2 -hard problem, 119
natural neighbor interpolant, 44
nearest neighbor problem, 183, 258
neighborly polytope, 251
nerve of a set of balls, 262
network Voronoi diagram, 228, 246
Newton’s method, 243
nice curve, 145, 168
nice metric, 146
norm, 130
NP-hard problem, 45, 91, 93, 103,
139, 197, 198, 212, 234, 238
offset of a shape, 57, 74, 221
on-line algorithm, 109
on-line clustering, 211
one-round Voronoi game, 234
Index
online algorithm, 22
open set, 8
optimal clustering, 191, 211, 212
orbit of a point, 126
order-k abstract Voronoi diagram,
174
order-k Delaunay graph, 207, 249
order-k Delaunay triangulation, 208
order-k Gabriel graph, 209
order-k power diagram, 82, 106
order-k segment Voronoi diagram, 52
order-k Voronoi diagram, 44, 85, 103,
185, 209, 214, 226
oriented sphere, 165
orphan-free Voronoi diagram, 150,
159, 164
orthogonal dual, 39
orthogonal polygon, 59
orthogonal polyhedron, 61
orthogonal sphere, 88
orthogonal walk, 80
outerplanar graph, 210, 211
output-sensitive algorithm, 85, 108,
201
outward graph Voronoi diagram, 247
overlay of convex partitions, 102
packing, 238
parallel algorithm, 27, 51, 91, 113,
245, 268
parametric search, 66
partial least-squares matching, 101
partial order, 22, 45
partition of a set, 9, 112
partition theorems (power diagram),
96
path planning, 216, 248
path-connected, 150, 169, 177
pattern matching, 101
pattern recognition, 120, 200, 204
peeper’s Voronoi diagram, 64
perfect matching, 198, 237
perturbation scheme, 270
pixel Voronoi diagram, 15, 33, 45,
162, 268
planar graph, 8, 15, 51, 134, 170, 179,
246, 249
planar straight line graph, 8, 47
Index
plane sweep, 28, 50, 63, 66, 70, 112,
136, 152, 155, 173, 186, 188
Plateau’s laws, 161
plesiohedron, 129
Poincaré model of hyperbolic
geometry, 124
point-location, 20, 80, 111, 184, 218,
226, 261
point-location in equal balls, 260
point-symmetric region, 229, 234
point-symmetric set, 130, 139
Poisson Voronoi diagram, 4
polar Voronoi diagram, 219
polarity, 84, 110
pole vertex, 201
polygonal surface, 58, 92, 127
polyhedral cell complex, 75, 82, 87
polyhedral distance function, 115, 142
polyhedron, 83
polylog function, 128
polynomial root finding, 242
polynomial-time algorithm, 45, 93,
203, 209, 212
polynomiography, 244
polytope, 84, 89, 93, 251
post office problem, 183, 220, 226,
228, 248, 258
potential-field method, 223
power cell, 81
power diagram, 40, 81, 94, 117, 152,
201, 213, 264
power function, 81, 94
pre-triangulation, 92
preorder of a tree, 70, 198
Prim’s algorithm, 196
priority queue, 30, 105, 189, 227
prototile of a lattice, 129
proximity problem, 186, 258
pseudo-circles, 133
pseudo-lines, 250
pseudo-parabolas, 250
pseudo-simplicial complex, 92
pseudo-triangulation, 58, 92, 217
quad-edge data structure, 15, 26, 31
quadrangle inequality, 189
quadratic-form distance, 129, 165
quadtree, 27, 261
335
quasi-Euclidean distance, 165
quickest path, 176
quotient space of a surface, 126
r-sample, 201
radical axis of circles, 81
radii graph, 199
random sampling, 226
randomized algorithm, 20, 22, 50, 51,
53, 66, 71, 109, 152, 172, 181, 193,
225
reciprocal figure, 39
recognition of shape Delaunay, 139
recognition of Voronoi diagrams, 35
rectilinear polygon, 59
recursive algorithm, 24
regular simplicial complex, 45, 88,
117, 264
regular tessellation, 88
regular triangulation, 44, 84, 201
regularizing flip, 91
relative neighborhood graph, 204
removing a Delaunay site, 106
restricted Delaunay complex, 263
retraction approach, 218
Riemannian manifold, 127
Rips complex, 262
roadmap, 222
robot motion planning, 216
roof construction, 59
root finding, 242
rotational motion planning, 220
roughness of triangular surface, 44
roundness of point set, 194
sampling technique, 116, 201, 222,
261
scene analysis, 40
Schönhard polytope, 93
Schauder’s fixed-point theorem, 241
Schlegel diagram, 83
scope of a triangle, 207
search tree, 22, 27, 30, 225, 256
second-closest pair, 189
secondary polytope, 90, 93
segment Delaunay triangulation, 51
self-centered power diagram, 264
self-centered tetrahedrization, 76
336
self-centered Voronoi diagram, 244
self-edge of a site, 67
semi-separated pair decomposition,
257
sensor network, 199, 248
separating regions problem, 236
separator, 7, 168
shape Delaunay tessellation, 136, 205,
228
shape reconstruction, 117, 200, 201
sharp P-hard, 129
shelling order, 45, 106
shortest connection network, 195, 197
shortest path, 64, 123, 144, 147, 162,
175, 205, 220, 246
similarity problem, 258
simple cell complex, 82, 86, 87
simple curve, 10, 145
simple polygon, 48, 50, 52, 57, 63, 191
simplex, 87
simplicial cell complex, 87
simply connected, 49, 57, 115, 125,
150, 154, 171, 177
single-linkage clustering, 215
singular face, 117
site event (plane sweep), 28
skeleton, 48, 55
skew distance, 156
skew Voronoi diagram, 156
skin surface for balls, 119
slab method for point-location, 184
sliver simplex, 263
smallest annulus problem, 193
smallest enclosing circle, 43, 192, 213
smallest enclosing shape, 139
smallest enclosing sphere, 192, 213
smooth object, 133, 136, 221, 222
Snell’s law of refraction, 175, 222
soap bubbles, 161
space-filler, 129
spanner of a point set, 205, 257, 260,
265
spanning cycle, 198, 199
spanning ratio of a graph, 205
spanning tree, 195
sparse graph, 205
spatial sampling, 116
sphere packing, 191
Index
spike event (plane sweep), 29
spiral search, 23
spline curve, 70
split event (straight skeleton), 56
split tree, 256
spread of a finite point set, 252
stable Delaunay subgraph, 228
star-shaped region, 106, 120, 133,
154, 159, 231
star-shapedness in a graph, 247
starting tetrahedron, 79
starting triangle, 21, 23, 111
Steinitz’s theorem, 92
stochastic properties of diagrams, 267
straight skeleton, 53, 55, 162, 177
straight skeleton for pixel shapes, 61
straight walk, 80
stretch factor of a graph, 205
strict triangle inequality, 133
strictly convex set, 131, 136
subgraph of the Delaunay, 194
supergraph of the Delaunay, 207
support hull, 137
surface reconstruction, 128, 201, 261
surface sampling, 5, 116, 264
sweep-circle algorithm, 125
sweep-line algorithm, 28
symmetric axis, 48, 71
symmetric distance function, 130,
139, 144
tentative prune-and-search, 186
terrain modeling, 43, 58, 208
terrain recognition, 40
territory diagram, 241
tessellation, 11
Thales’ theorem, 157, 227
Thiessen polygons, 1
tilted plane, 155
top-down divide & conquer, 27, 67
topological clustering, 216
torus, 127, 231
tracing method, 116
translational motion planning, 220
transportation network, 176, 248
transportation problem, 100, 155
travelling salesman tour, 198
Index
tree, 10, 47, 50, 53, 57, 67, 174, 210,
211, 246
triangle inequality, 130, 144
triangular network, 11, 41
triangular surface, 44, 110, 208
triangulation, 11, 41, 47
triangulation axis of polygon, 55
trisector curve, 239
two-center problem, 193
two-site Voronoi diagram, 113
unavoidable triangulation edge, 203
unbounded set, 8
uniform triangular mesh, 191
union of balls, 82, 88, 117, 201
union-find data structure, 196
unit-disk graph, 248
untangling a polygon, 41
upper bound theorem, 83
upper envelope of (hyper)planes, 83,
106, 110
upper envelope of surfaces, 112, 160
variance of a cluster, 100, 212
vertex-removing flip, 90
vertical projection, 32, 83, 84
very nice metric, 149, 171
Vietoris–Rips complex, 262
visibility graph, 220
visibility walk, 80
visibility-constrained Voronoi
diagram, 64, 162, 222
visibility-Voronoi complex, 221
visible, 49
Voronoi art, 244
Voronoi diagram, 8
Voronoi diagram for a lattice, 129
Voronoi diagram for circles, 66, 145,
154
Voronoi diagram for curved objects,
66
Voronoi diagram for graphs, 228, 246
Voronoi diagram for line segments,
47, 114, 143, 185, 218
Voronoi diagram for lines, 114, 143
Voronoi diagram for moving sites, 27,
50, 136, 227
Voronoi diagram for obstacles, 64, 176
337
Voronoi diagram for oriented spheres,
165
Voronoi diagram for periodical sites,
129
Voronoi diagram for spheres, 86, 120
Voronoi diagram in 3-space, 75
Voronoi diagram in a river, 156
Voronoi diagram inversion, 35
Voronoi diagram of a convex n-gon,
54
Voronoi diagram on a cone, 124
Voronoi diagram on a torus, 128, 231
Voronoi diagram on polyhedra, 127
Voronoi diagram on the sphere, 123
Voronoi diagrams on surfaces, 123
Voronoi diagrams on the grid, 268
Voronoi edge, 8, 75
Voronoi facet, 75
Voronoi game, 233, 238
Voronoi region, 7
Voronoi surface, 33, 112, 114, 168, 271
Voronoi tree, 27
Voronoi tree map, 60
Voronoi vertex, 8, 75
Walsh matrix, 254
wavefront, 28, 55, 65, 177
wavefront model, 9, 144, 161
weighted α-shape, 88, 117, 201
weighted centroid, 44, 244
weighted region problem, 175
weighted skeleton, 60, 96
weighted Voronoi diagram, 86, 152,
228
well-separated pair decomposition,
255
well-separated point sets, 255
width of a point set, 119
Wigner–Seitz zones, 2
Wirkungsbereiche, 2
witness complex, 210, 263
witness Delaunay graph, 210, 236,
248, 265
witness Gabriel graph, 211
zone diagram, 238
zone in an arrangement, 114
Zorn’s lemma, 242